\displaystyle \text{MULTPLE CHOICE QUESTIONS (MCQ)}

\displaystyle \textbf{Question 1: }\text{The set of intelligent students in a class is:}
\displaystyle \text{(a) A null set}\qquad\text{(b) A singleton set}
\displaystyle \text{(c) A finite set}\qquad\text{(d) Not a well defined collection}
\displaystyle \text{Answer:}
\displaystyle \text{The term ``intelligent'' is subjective and does not give a definite criterion for deciding}
\displaystyle \text{whether a student belongs to the collection or not. Hence, it is not a well defined collection.}
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{If the sets }A\text{ and }B\text{ are given by }A=\{1,2,3,4\},\ B=\{2,4,6,8,10\}
\displaystyle \text{and the universal set }U=\{1,2,3,4,5,6,7,8,9,10\},\text{ then:}
\displaystyle \text{(a) }(A\cup B)'=\{5,7,9\}
\displaystyle \text{(b) }(A\cap B)'=\{1,3,5,6,7\}
\displaystyle \text{(c) }(A\cap B)'=\{1,3,5,6,7,8\}
\displaystyle \text{(d) None of these}
\displaystyle \text{Answer:}
\displaystyle A\cup B=\{1,2,3,4,6,8,10\}.
\displaystyle \therefore (A\cup B)'=U-(A\cup B)=\{5,7,9\}.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{If }A=\{1,2,3,4\},\ B=\{2,3,5,6\}\text{ and }C=\{3,4,6,7\},\text{ then}
\displaystyle \text{(a) }A-(B\cap C)=\{1,3,4\}\qquad\text{(b) }A-(B\cap C)=\{1,2,4\}
\displaystyle \text{(c) }A-(B\cup C)=\{2,3\}\qquad\text{(d) }A-(B\cup C)=\{\emptyset\}
\displaystyle \text{Answer:}
\displaystyle B\cap C=\{3,6\}.
\displaystyle \therefore A-(B\cap C)=\{1,2,3,4\}-\{3,6\}=\{1,2,4\}.
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{The set }\{x:x\text{ is an even prime number}\}\text{ can be written as:}
\displaystyle \text{(a) }\{2\}\qquad\text{(b) }\{2,4\}\qquad\text{(c) }\{2,14\}\qquad\text{(d) }\{2,4,14\}
\displaystyle \text{Answer:}
\displaystyle 2\text{ is the only even prime number.}
\displaystyle \therefore \{x:x\text{ is an even prime number}\}=\{2\}.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{The number of the proper subset of }\{a,b,c\}\text{ is:}
\displaystyle \text{(a) }3\qquad\text{(b) }8\qquad\text{(c) }6\qquad\text{(d) }7
\displaystyle \text{Answer:}
\displaystyle \text{A set containing }n\text{ elements has }2^n\text{ subsets.}
\displaystyle \text{Here, }n=3.\text{ Therefore, the number of subsets is }2^3=8.
\displaystyle \text{A proper subset cannot be equal to the given set itself.}
\displaystyle \therefore \text{Number of proper subsets}=8-1=7.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Which one is different from the others?}
\displaystyle \text{(i) empty set}\qquad\text{(ii) void set}\qquad\text{(iii) zero set}\qquad\text{(iv) null set}
\displaystyle \text{(a) (i)}\qquad\text{(b) (ii)}\qquad\text{(c) (iii)}\qquad\text{(d) (iv)}
\displaystyle \text{Answer:}
\displaystyle \text{Empty set, void set and null set are different names for a set having no elements.}
\displaystyle \text{A zero set is not a standard synonym for the empty set. Hence, it is different from the others.}
\displaystyle \therefore \text{The correct option is (c), i.e., (iii).}
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Given the sets }A=\{1,3,5\},\ B=\{2,4,6\}\text{ and }C=\{0,2,4,6,8\}.\text{ Which of the}
\displaystyle \text{following may be considered as universal set for all the three sets }A,\ B\text{ and }C\text{:}
\displaystyle \text{(a) }\{0,1,2,3,4,5,6\}\qquad\text{(b) }\emptyset
\displaystyle \text{(c) }\{0,1,2,3,4,5,6,7,8,9,10\}\qquad\text{(d) }\{1,2,3,4,5,6,7,8\}
\displaystyle \text{Answer:}
\displaystyle \text{A universal set must contain every element of }A,\ B\text{ and }C.
\displaystyle A\cup B\cup C=\{0,1,2,3,4,5,6,8\}.
\displaystyle \text{Only option (c) contains all these elements.}
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Which of the following collections is a set?}
\displaystyle \text{(a) The collection of all the days of a week}
\displaystyle \text{(b) A collection of }11\text{ best hockey player of India.}
\displaystyle \text{(c) The collection of all rich person of Delhi}
\displaystyle \text{(d) A collection of most dangerous animals of India.}
\displaystyle \text{Answer:}
\displaystyle \text{A set is a well defined collection of distinct objects.}
\displaystyle \text{The collection of all the days of a week is well defined because its members can be identified}
\displaystyle \text{without any ambiguity. The terms ``best'', ``rich'' and ``most dangerous'' are subjective.}
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{If }A\cup B=\emptyset\text{ then }n(A\cup B)\text{ is equal to:}
\displaystyle \text{(a) }n(A)+n(B)-n(A\cap B)\qquad\text{(b) }n(A)-n(B)+n(A\cap B)
\displaystyle \text{(c) }n(A)+n(B)+n(A\cap B)\qquad\text{(d) }n(A)-n(B)-n(A\cap B)
\displaystyle \text{Answer:}
\displaystyle \text{For any two finite sets }A\text{ and }B,
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B).
\displaystyle \text{Also, }A\cup B=\emptyset\text{ implies }A=B=\emptyset,\text{ so both sides are equal to }0.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Let }V=\{a,e,i,o,u\}\text{ and }B=\{a,i,k,u\}\text{ then value of }
\displaystyle (V-B)\text{ and }(B-V) \ \text{are respectively:}
\displaystyle \text{(a) }\{e,o\}\text{ and }\{k\}\qquad\text{(b) }\{e\}\text{ and }\{k\}
\displaystyle \text{(c) }\{o\}\text{ and }\{k\}\qquad\text{(d) }\{e,o\}\text{ and }\{k,i\}
\displaystyle \text{Answer:}
\displaystyle V-B=\{e,o\},\text{ since }e\text{ and }o\text{ are the elements of }V\text{ which are not in }B.
\displaystyle B-V=\{k\},\text{ since }k\text{ is the only element of }B\text{ which is not in }V.
\displaystyle \therefore (V-B)=\{e,o\}\text{ and }(B-V)=\{k\}.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Let }A=\{a,b\},\ B=\{a,b,c\}\text{ then }A\cup B\text{ is:}
\displaystyle \text{(a) }\{a,b\}\qquad\text{(b) }\{a,c\}\qquad\text{(c) }\{a,b,c\}\qquad\text{(d) }\{b,c\}
\displaystyle \text{Answer:}
\displaystyle A\cup B\text{ contains all the elements which belong to }A\text{ or }B\text{ or both.}
\displaystyle \therefore A\cup B=\{a,b,c\}.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{The number of subsets of a set containing }n\text{ elements is:}
\displaystyle \text{(a) }n\qquad\text{(b) }2^{n-1}\qquad\text{(c) }n^2\qquad\text{(d) }2^n
\displaystyle \text{Answer:}
\displaystyle \text{For each of the }n\text{ elements, there are two choices: either include it in a subset or exclude it.}
\displaystyle \therefore \text{The total number of subsets}=2\times2\times\cdots\times2=2^n.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

 

\displaystyle \textbf{Question 13: }\text{If }A=\{1,2,3\},\ B=\{1,4,6,9\},\text{ and }R\text{ is a relation from }A\text{ to }B\text{ defined}
\displaystyle \text{by `}x\text{ is greater than }y\text{' the range of }R\text{ is:}
\displaystyle \text{(a) }\{1,4,6,9\}\qquad\text{(b) }\{4,6,9\}\qquad\text{(c) }\{1\}\qquad\text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle R=\{(x,y):x\in A,\ y\in B\text{ and }x>y\}.
\displaystyle \text{Since }A=\{1,2,3\}\text{ and }B=\{1,4,6,9\},\text{ the possible ordered pairs are}
\displaystyle R=\{(2,1),(3,1)\}.
\displaystyle \text{The range of }R\text{ is the set of second elements of the ordered pairs.}
\displaystyle \therefore \text{Range of }R=\{1\}.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{Let }R\text{ be a relation from a set }A\text{ to set }B\text{ then:}
\displaystyle \text{(a) }R=A\cup B\qquad\text{(b) }R=A\cap B\qquad\text{(c) }R\subseteq A\times B\qquad\text{(d) }R\subseteq B\times A
\displaystyle \text{Answer:}
\displaystyle \text{By definition, a relation }R\text{ from }A\text{ to }B\text{ is any subset of the Cartesian product }A\times B.
\displaystyle \therefore R\subseteq A\times B.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{The number of subsets of a set containing }n\text{ elements is:}
\displaystyle \text{(a) }n\qquad\text{(b) }2^{n-1}\qquad\text{(c) }n^2\qquad\text{(d) }2^n
\displaystyle \text{Answer:}
\displaystyle \text{For each of the }n\text{ elements, there are two choices: either include it in a subset or exclude it.}
\displaystyle \therefore \text{The total number of subsets}=2^n.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{If }A=\{1,2,3\},\ B=\{1,4,6,9\},\text{ and }R\text{ is a relation from }A\text{ to }B\text{ defined by}
\displaystyle \text{`x is greater than y' the range of }R\text{ is:}
\displaystyle \text{(a) }\{1,4,6,9\}\qquad\text{(b) }\{4,6,9\}\qquad\text{(c) }\{1\}\qquad\text{(d) None of these}
\displaystyle \text{Answer:}
\displaystyle R=\{(x,y):x\in A,\ y\in B\text{ and }x>y\}.
\displaystyle \text{The ordered pairs satisfying the given condition are }(2,1)\text{ and }(3,1).
\displaystyle \therefore R=\{(2,1),(3,1)\}.
\displaystyle \text{The range of }R\text{ is the set of second elements of the ordered pairs.}
\displaystyle \therefore \text{Range of }R=\{1\}.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{Set }A=\{0,7,26,63\}\text{ in set-builder form is:}
\displaystyle \text{(a) }\{x:x\in N,\ x=n^3-1\text{ and }n<7\}
\displaystyle \text{(b) }\{x:x\in N,\ x=n^2-1\text{ and }n<5\}
\displaystyle \text{(c) }\{x:x\in N,\ x=n^2-1\text{ and }n<4\}
\displaystyle \text{(d) }\{x:x\in N,\ x=n^3-1\text{ and }n\leq4\}
\displaystyle \text{Answer:}
\displaystyle 0=1^3-1,\qquad 7=2^3-1,\qquad 26=3^3-1,\qquad 63=4^3-1.
\displaystyle \therefore A=\{x:x=n^3-1,\ n\in N,\ n\leq4\}.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{Let }A,\ B,\ C\text{ be three sets. If }A\in B\text{ and }B\subset C,\text{ then:}
\displaystyle \text{(a) }A\subset C\qquad\text{(b) }A\notin C\qquad\text{(c) }A\in C\qquad\text{(d) }A\not\subset C
\displaystyle \text{Answer:}
\displaystyle \text{Given }A\in B,\text{ so }A\text{ is an element of }B.
\displaystyle \text{Also, }B\subset C,\text{ so every element of }B\text{ is also an element of }C.
\displaystyle \therefore A\in C.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{If }B=\{\emptyset\}\text{ then:}
\displaystyle \text{(a) }B\text{ is an empty set}\qquad\text{(b) }B\text{ is a finite set}
\displaystyle \text{(c) }B\text{ is an infinite set}\qquad\text{(d) }B\text{ is not a set}
\displaystyle \text{Answer:}
\displaystyle B=\{\emptyset\}\text{ contains one element, namely the empty set }\emptyset.
\displaystyle \therefore n(B)=1.
\displaystyle \text{Hence, }B\text{ is a finite set and not an empty set.}
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{If }A=\{3,5,7,9,11\},\ B=\{7,9,11,13\},\ C=\{11,13,15\}\text{ and }
\displaystyle D=\{15,17\}, \ \text{then }(A\cup B)\cap C\text{ is:}
\displaystyle \text{(a) }\{11,13,15\}\qquad\text{(b) }\{9,11,13,15\}
\displaystyle \text{(c) }\{9,11,13\}\qquad\text{(d) }\{11,13\}
\displaystyle \text{Answer:}
\displaystyle A\cup B=\{3,5,7,9,11,13\}.
\displaystyle \therefore (A\cup B)\cap C=\{3,5,7,9,11,13\}\cap\{11,13,15\}.
\displaystyle \therefore (A\cup B)\cap C=\{11,13\}.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 21: }\text{If }n(A)=2\text{ then }n[P(P(A))]\text{ is:}
\displaystyle \text{(a) }8\qquad\text{(b) }4\qquad\text{(c) }12\qquad\text{(d) }16
\displaystyle \text{Answer:}
\displaystyle \text{Since }n(A)=2,\text{ the number of elements in the power set of }A\text{ is}
\displaystyle n[P(A)]=2^{n(A)}=2^2=4.
\displaystyle \text{Therefore, the number of elements in the power set of }P(A)\text{ is}
\displaystyle n[P(P(A))]=2^{n[P(A)]}=2^4=16.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 22: }\text{Let }P\text{ be a set of squares, }Q\text{ be set of parallelograms, }R\text{ be a set of}
\displaystyle \text{quadrilaterals and }S\text{ be a set of rectangles. Consider the following:}
\displaystyle \text{(I) }P\subset Q\qquad\text{(II) }R\subset P\qquad\text{(III) }P\subset S\qquad\text{(IV) }S\subset R
\displaystyle \text{Which of the above are correct?}
\displaystyle \text{(a) I, II and III}\qquad\text{(b) I, III and IV}
\displaystyle \text{(c) I, II and IV}\qquad\text{(d) III and IV}
\displaystyle \text{Answer:}
\displaystyle \text{Every square is a parallelogram. Therefore, }P\subset Q\text{ is true.}
\displaystyle \text{Every quadrilateral need not be a square. Therefore, }R\subset P\text{ is false.}
\displaystyle \text{Every square is a rectangle. Therefore, }P\subset S\text{ is true.}
\displaystyle \text{Every rectangle is a quadrilateral. Therefore, }S\subset R\text{ is true.}
\displaystyle \therefore \text{Statements I, III and IV are correct.}
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 23: }A=\{1,2,3,4\},\ B=\{-1,1,0,-2,2\},\ C=\{1,3,4\}\text{ are subset of which set?}
\displaystyle \text{(a) }[1,4]\qquad\text{(b) }[-1,4]\qquad\text{(c) }[-2,2]\qquad\text{(d) }[-2,4]
\displaystyle \text{Answer:}
\displaystyle \text{For a set to contain }A,\ B\text{ and }C\text{ as subsets, it must contain all their elements.}
\displaystyle A\cup B\cup C=\{-2,-1,0,1,2,3,4\}.
\displaystyle \text{All these elements belong to the interval }[-2,4].
\displaystyle \therefore A\subset[-2,4],\quad B\subset[-2,4],\quad C\subset[-2,4].
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 24: }\text{Let }A=\{(1,2),(3,4),5\},\text{ then which of the following is incorrect?}
\displaystyle \text{(a) }\{3,4\}\notin A\text{ as }(3,4)\text{ is an element of }A
\displaystyle \text{(b) }\{5\},\{(3,4)\}\text{ are subsets of }A\text{ but not elements of }A
\displaystyle \text{(c) }\{1,2\},\{5\}\text{ are subsets of }A
\displaystyle \text{(d) }\{(1,2),(3,4),5\}\text{ is a subset of }A
\displaystyle \text{Answer:}
\displaystyle A=\{(1,2),(3,4),5\}\text{ has three elements, namely }(1,2),(3,4)\text{ and }5.
\displaystyle \text{Since }1\notin A\text{ and }2\notin A,\text{ the set }\{1,2\}\text{ is not a subset of }A.
\displaystyle \text{However, }5\in A,\text{ so }\{5\}\subset A.
\displaystyle \therefore \text{Statement (c) is incorrect.}
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 25: }\text{Match the following sets in column I with the intervals in column II:}
\displaystyle \begin{array}{c|l|c|c} &\text{Column-I}&&\text{Column-II}\\ \hline A&\{x:x\in R,\ a<x<b\}&&1.\ (a,b]\\ B&\{x\in R:a\leq x\leq b\}&&2.\ [a,b)\\ C&\text{The set of real numbers }x\text{ such that }a\leq x<b&&3.\ (a,b)\\ D&\{x:x\in R\text{ and }a<x\leq b\}&&4.\ [a,b] \end{array}

\displaystyle \text{Codes:}
\displaystyle \begin{array}{c|cccc} &A&B&C&D\\ \hline \text{(a)}&4&1&2&3\\ \text{(b)}&2&3&4&1\\ \text{(c)}&1&2&3&4\\ \text{(d)}&3&4&2&1 \end{array}

\displaystyle \text{Answer:}
\displaystyle A=\{x:x\in R,\ a<x<b\}=(a,b).
\displaystyle \therefore A\rightarrow3.
\displaystyle B=\{x\in R:a\leq x\leq b\}=[a,b].
\displaystyle \therefore B\rightarrow4.
\displaystyle C=\{x\in R:a\leq x<b\}=[a,b).
\displaystyle \therefore C\rightarrow2.
\displaystyle D=\{x:x\in R\text{ and }a<x\leq b\}=(a,b].
\displaystyle \therefore D\rightarrow1.
\displaystyle \therefore A\rightarrow3,\quad B\rightarrow4,\quad C\rightarrow2,\quad D\rightarrow1.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 26: }\text{Two finite sets have }m\text{ and }n\text{ elements. The number of elements in the power set}
\displaystyle \text{of first set is }48\text{ more than the total number of elements in power set of the second set. Then}
\displaystyle \text{the value of }m\text{ and }n\text{ are:}
\displaystyle \text{(A) }7,6\qquad\text{(B) }6,3\qquad\text{(C) }6,4\qquad\text{(D) }7,4
\displaystyle \text{Answer:}
\displaystyle \text{A set having }m\text{ elements has }2^m\text{ elements in its power set.}
\displaystyle \text{Similarly, a set having }n\text{ elements has }2^n\text{ elements in its power set.}
\displaystyle \text{According to the given condition,}
\displaystyle 2^m-2^n=48.
\displaystyle \text{Checking the given options, for }m=6\text{ and }n=4,
\displaystyle 2^6-2^4=64-16=48.
\displaystyle \therefore m=6,\quad n=4.
\displaystyle \therefore \text{The correct option is (C).}
\displaystyle \\

\displaystyle \textbf{Question 27: }\text{Which of the following is correct?}
\displaystyle \text{I. Number of non-empty subsets of a set having }n\text{ elements are }2^n-1.
\displaystyle \text{II. The number of non-empty subsets of the set }\{a,b,d\}\text{ are }15.
\displaystyle \text{(a) Only I is false}\qquad\text{(b) Only II is false}
\displaystyle \text{(c) Both I and II are false}\qquad\text{(d) Both I and II are true}
\displaystyle \text{Answer:}
\displaystyle \text{A set having }n\text{ elements has }2^n\text{ subsets, including the empty set.}
\displaystyle \therefore \text{Number of non-empty subsets}=2^n-1.
\displaystyle \text{Hence, statement I is true.}
\displaystyle \text{The set }\{a,b,d\}\text{ has }3\text{ elements. Therefore, the number of its non-empty subsets is}
\displaystyle 2^3-1=8-1=7.
\displaystyle \text{Hence, statement II is false.}
\displaystyle \therefore \text{Only statement II is false.}
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 28: }\text{Which of the following is not a null set?}
\displaystyle \text{(a) Set of odd natural numbers divisible by }2
\displaystyle \text{(b) Set of even prime numbers}
\displaystyle \text{(c) }\{x:x\text{ is a natural number, }x>5\text{ and }x<6\}
\displaystyle \text{(d) }\{y:y\text{ is a point common to any two parallel lines}\}
\displaystyle \text{Answer:}
\displaystyle \text{There is no odd natural number divisible by }2,\text{ so (a) represents a null set.}
\displaystyle \text{There is no natural number lying strictly between }5\text{ and }6,\text{ so (c) represents a null set.}
\displaystyle \text{Two distinct parallel lines have no common point, so (d) also represents a null set.}
\displaystyle \text{But }2\text{ is an even prime number. Hence, the set in (b) is }\{2\}\text{ and is not a null set.}
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 29: }\text{If }n(A)=4,\text{ how many proper subsets does the set }A\text{ have?}
\displaystyle \text{(a) }15\qquad\text{(b) }16\qquad\text{(c) }3\qquad\text{(d) }14
\displaystyle \text{Answer:}
\displaystyle \text{A set having }n\text{ elements has }2^n\text{ subsets.}
\displaystyle \therefore \text{Number of subsets of }A=2^4=16.
\displaystyle \text{A proper subset cannot be equal to the set itself.}
\displaystyle \therefore \text{Number of proper subsets}=16-1=15.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 30: }A\text{ and }B\text{ are not singleton sets and }n(A\times B)=21.
\displaystyle \text{ If }A\subset B\text{ then }n(B) \ \text{is equal to:}
\displaystyle \text{(a) }3\qquad\text{(b) }7\qquad\text{(c) }21\qquad\text{(d) }1
\displaystyle \text{Answer:}
\displaystyle n(A\times B)=n(A)\cdot n(B)=21.
\displaystyle \text{Since }21=3\times7\text{ and neither }A\text{ nor }B\text{ is a singleton set,}
\displaystyle n(A)=3,\qquad n(B)=7\quad\text{or}\quad n(A)=7,\qquad n(B)=3.
\displaystyle \text{But }A\subset B,\text{ so }n(A)<n(B).
\displaystyle \therefore n(A)=3\text{ and }n(B)=7.
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 31: }\text{Which of the following are disjoint sets:}
\displaystyle \text{(a) }\{a,b,c\}\text{ \& }\{b,c,a\}
\displaystyle \text{(b) }\{a,e,b,d\}\text{ \& }\{d,a,b,e\}
\displaystyle \text{(c) }\{2,5,m,n\}\text{ \& }\{n,m,5,2\}
\displaystyle \text{(d) }\{2,4,6,8\}\text{ \& }\{x:x\text{ is an even number}\}
\displaystyle \text{Answer:}
\displaystyle \text{Two sets are disjoint if they have no common element, i.e., their intersection is }\emptyset.
\displaystyle \text{In (a), both sets contain the same elements }a,b,c,\text{ so they are not disjoint.}
\displaystyle \text{In (b), both sets contain the same elements }a,b,d,e,\text{ so they are not disjoint.}
\displaystyle \text{In (c), both sets contain the same elements }2,5,m,n,\text{ so they are not disjoint.}
\displaystyle \text{In (d), }2,4,6,8\text{ are all even numbers, so the sets have common elements.}
\displaystyle \therefore \text{none of the given pairs are disjoint sets.}
\displaystyle \text{Hence, Question 31 has no correct option among the choices given.}
\displaystyle \\

\displaystyle \textbf{Question 32: }A\text{ and }B\text{ are non-empty sets, then }P(A)\cup P(B)\text{ is equal to:}
\displaystyle \text{(a) }P(A\cup B)\qquad\text{(b) }P(A\cap B)\qquad\text{(c) }P(A)=P(B)\qquad\text{(d) None of these}
\displaystyle \text{Answer:}
\displaystyle P(A)\cup P(B)\text{ consists of all subsets which are subsets of }A\text{ or subsets of }B.
\displaystyle \text{In general, }P(A)\cup P(B)\ne P(A\cup B),\text{ because }P(A\cup B)\text{ may contain subsets having}
\displaystyle \text{elements from both }A\text{ and }B.
\displaystyle \text{Also, }P(A)\cup P(B)\ne P(A\cap B)\text{ in general.}
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 33: }\text{Consider the sets }A=\{0\},\ B=\{x:x>15\text{ and }
\displaystyle x<5\},\ C=\{x:x-5=0\}, \ D=\{x:x^2=25\}\text{ and }
\displaystyle E=\{x:x\text{ is an integral positive} \ \text{root of the equation }x^2-2x-15=0\}.  \\ \text{Choose the pair of equal sets:}
\displaystyle \text{(a) }A\text{ and }B\qquad\text{(b) }C\text{ and }D\qquad\text{(c) }C\text{ and }E\qquad\text{(d) }B\text{ and }C
\displaystyle \text{Answer:}
\displaystyle C=\{x:x-5=0\}=\{5\}.
\displaystyle \text{For }E,\quad x^2-2x-15=0.
\displaystyle \therefore (x-5)(x+3)=0.
\displaystyle \therefore x=5\text{ or }x=-3.
\displaystyle \text{Since }E\text{ contains only the integral positive root, }E=\{5\}.
\displaystyle \therefore C=E=\{5\}.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 34: }\text{If }U=\{1,2,3,4,\ldots,10\}\text{ is the universal set of the sets }
\displaystyle A=\{2,4,6,8,10\} \ \text{and }B=\{4,6\},\text{ then given sets can be represented by Venn diagram as:}
\displaystyle \text{(a) Diagram (a)}\qquad\text{(b) Diagram (b)}\qquad\text{(c) Diagram (c)}\qquad\text{(d) Diagram (d)}
\displaystyle \text{Answer:}
\displaystyle A=\{2,4,6,8,10\}\text{ and }B=\{4,6\}.
\displaystyle \text{Every element of }B\text{ is also an element of }A.
\displaystyle \therefore B\subset A.
\displaystyle A-B=\{2,8,10\}.
\displaystyle U-A=\{1,3,5,7,9\}.
\displaystyle \text{Hence, the circle representing }B\text{ must lie completely inside the circle representing }A.
\displaystyle \text{The elements }4,6\text{ lie in }B,\text{ the elements }2,8,10\text{ lie in }A-B,
\displaystyle \text{and }1,3,5,7,9\text{ lie outside }A.
\displaystyle \therefore \text{The correct Venn diagram is (d).}
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 35: }\text{The shaded region in the given figure is} \displaystyle \text{(a) }B\cap(A\cup C)\qquad\text{(b) }B\cup(A\cap C)
\displaystyle \text{(c) }B\cap(A-C)\qquad\text{(d) }B-(A\cup C)
\displaystyle \text{Answer:}
\displaystyle \text{From the Venn diagram, the shaded region lies inside }B\text{ but outside both }A\text{ and }C.
\displaystyle \text{Therefore, the portions }B\cap A\text{ and }B\cap C\text{ are excluded from }B.
\displaystyle \therefore \text{Shaded region}=B-(A\cup C).
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 36: }\text{Which of the following sets is a finite set?}
\displaystyle \text{(a) }A=\{x:x\in Z\text{ and }x^2-5x+6=0\}
\displaystyle \text{(b) }B=\{x:x\in Z\text{ and }x^2\text{ is even}\}
\displaystyle \text{(c) }D=\{x:x\in Z\text{ and }x>-10\}
\displaystyle \text{(d) All of these}
\displaystyle \text{Answer:}
\displaystyle \text{For set }A,\quad x^2-5x+6=0.
\displaystyle \therefore (x-2)(x-3)=0.
\displaystyle \therefore x=2\text{ or }x=3.
\displaystyle \therefore A=\{2,3\},\text{ which is a finite set.}
\displaystyle \text{For set }B,\text{ every even integer has an even square, so }B\text{ has infinitely many elements.}
\displaystyle \text{Also, there are infinitely many integers greater than }-10,\text{ so }D\text{ is an infinite set.}
\displaystyle \therefore \text{Only }A\text{ is a finite set.}
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 37: }\text{Which of the following has only one subset:}
\displaystyle \text{(a) }\{\}\qquad\text{(b) }\{4\}\qquad\text{(c) }\{4,5\}\qquad\text{(d) }\{0\}
\displaystyle \text{Answer:}
\displaystyle \text{A set containing }n\text{ elements has }2^n\text{ subsets.}
\displaystyle \text{The empty set }\emptyset\text{ has }0\text{ elements.}
\displaystyle \therefore \text{Number of subsets of }\emptyset=2^0=1.
\displaystyle \text{Its only subset is }\emptyset\text{ itself.}
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 38: }\text{If }A\text{ and }B\text{ be any two sets, then }A\cap(A\cup B)'\text{ is equal to:}
\displaystyle \text{(a) }A\qquad\text{(b) }B\qquad\text{(c) }\emptyset\qquad\text{(d) None of these}
\displaystyle \text{Answer:}
\displaystyle \text{By De Morgan's law,}
\displaystyle (A\cup B)'=A'\cap B'.
\displaystyle \therefore A\cap(A\cup B)'=A\cap A'\cap B'.
\displaystyle \text{But }A\cap A'=\emptyset.
\displaystyle \therefore A\cap(A\cup B)'=\emptyset.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 39: }\text{A survey shows that }63\%\text{ of the people watch a news channel whereas }76\%
\displaystyle \text{watch another channel. If }x\%\text{ of the people watch both channel, then:}
\displaystyle \text{(a) }x=35\qquad\text{(b) }x=63\qquad\text{(c) }39\leq x\leq63\qquad\text{(d) }x=39
\displaystyle \text{Answer:}
\displaystyle \text{Let }A\text{ and }B\text{ represent the people watching the two news channels.}
\displaystyle n(A)=63,\qquad n(B)=76,\qquad n(A\cap B)=x.
\displaystyle \text{Since }n(A\cup B)\leq100,
\displaystyle 63+76-x\leq100.
\displaystyle \therefore 139-x\leq100.
\displaystyle \therefore x\geq39.
\displaystyle \text{Also, }n(A\cap B)\leq\min\{n(A),n(B)\}=63.
\displaystyle \therefore x\leq63.
\displaystyle \therefore 39\leq x\leq63.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 40: }A=\{x:x\ne x\}\text{ represents:}
\displaystyle \text{(a) }\{x\}\qquad\text{(b) }\{1\}\qquad\text{(c) }\{\}\qquad\text{(d) }\{0\}
\displaystyle \text{Answer:}
\displaystyle \text{For every element }x,\text{ we always have }x=x.
\displaystyle \text{Therefore, there is no element }x\text{ satisfying the condition }x\ne x.
\displaystyle \therefore A=\emptyset=\{\}.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 41: }\text{In a group of }52\text{ persons, }16\text{ drink tea but not coffee, while }33\text{ drink tea.}
\displaystyle \text{How many persons drink coffee but not tea:}
\displaystyle \text{(a) }17\qquad\text{(b) }36\qquad\text{(c) }23\qquad\text{(d) }19
\displaystyle \text{Answer:}
\displaystyle \text{Number of persons who drink both tea and coffee}=33-16=17.
\displaystyle \text{Assuming every person in the group drinks tea or coffee or both,}
\displaystyle \text{number of persons who drink coffee but not tea}=52-(16+17)=19.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 42: }\text{There are }600\text{ student in a school. If }400\text{ of them can speak Telugu, }300\text{ can}
\displaystyle \text{speak Hindi, then the number of students who can speak both Telugu and Hindi is:}
\displaystyle \text{(a) }100\qquad\text{(b) }200\qquad\text{(c) }300\qquad\text{(d) }400
\displaystyle \text{Answer:}
\displaystyle \text{Let }T\text{ and }H\text{ denote the sets of students who can speak Telugu and Hindi respectively.}
\displaystyle n(T)=400,\qquad n(H)=300,\qquad n(T\cup H)=600.
\displaystyle \text{Using }n(T\cup H)=n(T)+n(H)-n(T\cap H),
\displaystyle 600=400+300-n(T\cap H).
\displaystyle \therefore n(T\cap H)=700-600=100.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 43: }\text{Which one of the following is an infinite set:}
\displaystyle \text{(a) The set of human beings on the earth}
\displaystyle \text{(b) The set of water drops in a glass of water}
\displaystyle \text{(c) The set of trees in a forest}
\displaystyle \text{(d) The set of all primes}
\displaystyle \text{Answer:}
\displaystyle \text{The number of human beings on the earth, water drops in a glass and trees in a forest are finite.}
\displaystyle \text{However, there are infinitely many prime numbers.}
\displaystyle \therefore \text{The set of all primes is an infinite set.}
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 44: }\text{If }\emptyset\text{ denotes the empty set, then which one of the following is correct:}
\displaystyle \text{(a) }\emptyset\in\emptyset\qquad\text{(b) }\emptyset\in\{\emptyset\}\qquad\text{(c) }\{\emptyset\}\in\{\emptyset\}\qquad\text{(d) }0\in\emptyset
\displaystyle \text{Answer:}
\displaystyle \text{The set }\{\emptyset\}\text{ contains exactly one element, namely }\emptyset.
\displaystyle \therefore \emptyset\in\{\emptyset\}.
\displaystyle \text{Also, the empty set has no elements, so }\emptyset\notin\emptyset\text{ and }0\notin\emptyset.
\displaystyle \text{Further, }\{\emptyset\}\ne\emptyset,\text{ so }\{\emptyset\}\notin\{\emptyset\}.
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 45: }\text{If }B=\{x:x\text{ is a student presently studying in both classes X and XI}\}.
\displaystyle \text{ Then, the} \ \text{number of elements in set }B\text{ are:}
\displaystyle \text{(a) finite}\qquad\text{(b) infinite}\qquad\text{(c) zero}\qquad\text{(d) None of these}
\displaystyle \text{Answer:}
\displaystyle \text{A student cannot presently study in both class X and class XI at the same time.}
\displaystyle \therefore B=\emptyset.
\displaystyle \therefore n(B)=0.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 46: }\text{The set }\{x:x\text{ is a positive integer less than }6\text{ and }3^x-1\text{ is an even}
\displaystyle \text{number}\}\text{ in roster form is:}
\displaystyle \text{(a) }\{1,2,3,4,5\}\qquad\text{(b) }\{1,2,3,4,5,6\}
\displaystyle \text{(c) }\{2,4,6\}\qquad\text{(d) }\{1,3,5\}
\displaystyle \text{Answer:}
\displaystyle \text{The positive integers less than }6\text{ are }1,2,3,4,5.
\displaystyle \text{Since }3\text{ is odd, }3^x\text{ is odd for every positive integer }x.
\displaystyle \therefore 3^x-1\text{ is even for }x=1,2,3,4,5.
\displaystyle \therefore \text{The required set is }\{1,2,3,4,5\}.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 47: }\text{If }A\subset B\text{ and }A\ne B,\text{ then:}
\displaystyle \text{(a) }A\text{ is called a proper subset of }B\qquad\text{(b) }A\text{ is called a super set of }B
\displaystyle \text{(c) }A\text{ is not a subset of }B\qquad\text{(d) }B\text{ is a subset of }A
\displaystyle \text{Answer:}
\displaystyle \text{If every element of }A\text{ belongs to }B\text{ and }A\ne B,\text{ then }A\text{ is a proper subset of }B.
\displaystyle \therefore A\text{ is called a proper subset of }B.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 48: }\text{Which of the following is true:}
\displaystyle \text{(a) }a\in\{\{a\},b\}\qquad\text{(b) }\{b,c\}\subset\{a,\{b,c\}\}
\displaystyle \text{(c) }\{a,b\}\subset\{a,\{b,c\}\}\qquad\text{(d) None of these}
\displaystyle \text{Answer:}
\displaystyle \text{In (a), the elements of }\{\{a\},b\}\text{ are }\{a\}\text{ and }b,\text{ not }a.
\displaystyle \therefore a\notin\{\{a\},b\}.
\displaystyle \text{In (b), }b\text{ and }c\text{ are not individual elements of }\{a,\{b,c\}\}.
\displaystyle \therefore \{b,c\}\not\subset\{a,\{b,c\}\}.
\displaystyle \text{In (c), }b\notin\{a,\{b,c\}\}.
\displaystyle \therefore \{a,b\}\not\subset\{a,\{b,c\}\}.
\displaystyle \therefore \text{none of (a), (b) and (c) is true.}
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 49: }\text{The set of real numbers }\{x:a<x<b\}\text{ is called:}
\displaystyle \text{(a) open interval}\qquad\text{(b) closed interval}
\displaystyle \text{(c) semi-open interval}\qquad\text{(d) semi-closed interval}
\displaystyle \text{Answer:}
\displaystyle \{x:a<x<b\}=(a,b).
\displaystyle \text{Since neither }a\text{ nor }b\text{ is included, }(a,b)\text{ is an open interval.}
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 50: }\text{The set of all letters of the word `SCHOOL' is represented by:}
\displaystyle \text{I. }\{S,C,H,O,O,L\}\qquad\text{II. }\{S,C,H,O,L\}
\displaystyle \text{III. }\{C,H,L,O,S\}\qquad\text{IV. }\{S,C,H,L\}
\displaystyle \text{The correct code is:}
\displaystyle \text{(a) I and II}\qquad\text{(b) I, II and III}\qquad\text{(c) II and III}\qquad\text{(d) I, II, III and IV}
\displaystyle \text{Answer:}
\displaystyle \text{In a set, repetition of an element does not change the set, and the order of elements is immaterial.}
\displaystyle \text{The distinct letters of `SCHOOL' are }S,C,H,O,L.
\displaystyle \therefore \{S,C,H,O,O,L\}=\{S,C,H,O,L\}=\{C,H,L,O,S\}.
\displaystyle \text{Thus, I, II and III represent the same set of letters of the word `SCHOOL'.}
\displaystyle \text{Statement IV is incorrect because it does not contain }O.
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 51: }\text{The empty set is represented by:}
\displaystyle \text{I. }\emptyset\qquad\text{II. }\{\emptyset\}\qquad\text{III. }\{\}\qquad\text{IV. }\{\{\}\}
\displaystyle \text{(a) I and II}\qquad\text{(b) I and III}\qquad\text{(c) II and III}\qquad\text{(d) I and IV}
\displaystyle \text{Answer:}
\displaystyle \emptyset\text{ and }\{\}\text{ both represent a set containing no elements.}
\displaystyle \text{However, }\{\emptyset\}\text{ and }\{\{\}\}\text{ each contain one element, namely the empty set.}
\displaystyle \therefore \text{I and III represent the empty set.}
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 52: }\text{Which of the following is correct?}
\displaystyle \text{I. Number of non-empty subsets of a set having }n\text{ elements are }2^n-1.
\displaystyle \text{II. The number of non-empty subsets of the set }\{a,b,c,d\}\text{ are }15.
\displaystyle \text{(a) Only I is false}\qquad\text{(b) Only II is false}
\displaystyle \text{(c) Both I and II are false}\qquad\text{(d) Both I and II are true}
\displaystyle \text{Answer:}
\displaystyle \text{A set having }n\text{ elements has }2^n\text{ subsets, including the empty set.}
\displaystyle \therefore \text{Number of non-empty subsets}=2^n-1.
\displaystyle \text{Hence, statement I is true.}
\displaystyle \text{The set }\{a,b,c,d\}\text{ has }4\text{ elements.}
\displaystyle \therefore \text{Number of non-empty subsets}=2^4-1=16-1=15.
\displaystyle \text{Hence, statement II is also true.}
\displaystyle \therefore \text{Both I and II are true.}
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 53: }\text{Let }A=\{(1,2),(3,4),5\},\text{ then which of the following is incorrect:}
\displaystyle \text{(a) }\{3,4\}\notin A\text{ as }(3,4)\text{ is an element of }A
\displaystyle \text{(b) }\{5\},\{(3,4)\}\text{ are subsets of }A\text{ but not elements of }A
\displaystyle \text{(c) }\{1,2\},\{5\}\text{ are subsets of }A
\displaystyle \text{(d) }\{(1,2),(3,4),5\}\text{ is a subset of }A
\displaystyle \text{Answer:}
\displaystyle \text{The elements of }A\text{ are }(1,2),(3,4)\text{ and }5.
\displaystyle \text{Since }1\notin A\text{ and }2\notin A,\text{ the set }\{1,2\}\text{ is not a subset of }A.
\displaystyle \text{However, }5\in A,\text{ so }\{5\}\subset A.
\displaystyle \therefore \text{Statement (c) is incorrect.}
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 54: }\text{From }50\text{ students taking examination in Mathematics, Physics and Chemistry, each}
\displaystyle \text{of the students has passed in at least one of the subject, }37\text{ passed Mathematics, }24\text{ Physics}
\displaystyle \text{and }43\text{ Chemistry. Atmost }19\text{ passed Mathematics and Physics, atmost }29\text{ Mathematics}
\displaystyle \text{and Chemistry and atmost }20\text{ Physics and Chemistry. Then, the largest number that could}
\displaystyle \text{have passed all three examinations, are:}
\displaystyle \text{(a) }12\qquad\text{(b) }14\qquad\text{(c) }15\qquad\text{(d) }16
\displaystyle \text{Answer:}
\displaystyle \text{Let }M,\ P\text{ and }C\text{ denote the sets of students passing Mathematics, Physics and Chemistry.}
\displaystyle n(M)=37,\qquad n(P)=24,\qquad n(C)=43,\qquad n(M\cup P\cup C)=50.
\displaystyle \text{Let }n(M\cap P\cap C)=x.
\displaystyle \text{By the principle of inclusion-exclusion,}
\displaystyle 50=37+24+43-n(M\cap P)-n(M\cap C)-n(P\cap C)+x.
\displaystyle \therefore n(M\cap P)+n(M\cap C)+n(P\cap C)=54+x.
\displaystyle \text{But the maximum possible value of the left-hand side is}
\displaystyle 19+29+20=68.
\displaystyle \therefore 54+x\leq68.
\displaystyle \therefore x\leq14.
\displaystyle \text{Hence, the largest possible number of students passing all three examinations is }14.
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 55: }\text{Let }U\text{ be the set of all boys and girls in school. }G\text{ be the set of all girls in}
\displaystyle \text{the school. }B\text{ be the set of all boys in the school and }S\text{ be the set of all students in the}
\displaystyle \text{school who take swimming. Some but not all students in the school take swimming:} \displaystyle \text{(a) Diagram (a)}\qquad\text{(b) Diagram (b)}\qquad\text{(c) Diagram (c)}\qquad\text{(d) None of these}
\displaystyle \text{Answer:}
\displaystyle \text{No student can be both a boy and a girl. Therefore, }B\cap G=\emptyset.
\displaystyle \text{The set }S\text{ represents students who take swimming and may contain both boys and girls.}
\displaystyle \text{Since only some students take swimming, }S\text{ is not equal to the universal set }U.
\displaystyle \text{Thus, }B\text{ and }G\text{ must be disjoint, while }S\text{ may intersect both }B\text{ and }G.
\displaystyle \text{Diagram (c) correctly represents these relationships.}
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \text{ASSERTION \& REASON TYPE QUESTIONS}

\displaystyle \text{Directions : Each of these questions contains two statements, Assertion and Reason.}
\displaystyle \text{Each of these questions also has four alternative choices, only one of which is the correct}
\displaystyle \text{answer. You have to select one of the codes (a), (b), (c) and (d) given below.}

\displaystyle \textbf{Question 56: }\text{Assertion: The number of non-empty subsets of the set }\{a,b,c,d\}\text{ are }15.
\displaystyle \text{Reason: Number of non-empty subsets of a set having }n\text{ elements are }2^n-1.
\displaystyle \text{(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.}
\displaystyle \text{(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.}
\displaystyle \text{(c) Assertion is correct, reason is incorrect.}
\displaystyle \text{(d) Assertion is incorrect, reason is correct.}
\displaystyle \text{Answer:}
\displaystyle \text{The set }\{a,b,c,d\}\text{ contains }4\text{ elements.}
\displaystyle \text{Number of non-empty subsets of a set having }n\text{ elements}=2^n-1.
\displaystyle \therefore \text{Number of non-empty subsets}=2^4-1=16-1=15.
\displaystyle \text{Thus, the Assertion is correct and the Reason is also correct.}
\displaystyle \text{The Reason correctly explains the Assertion.}
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 56: }\text{Assertion: The number of non-empty subsets of the set }\{a,b,c,d\}\text{ are }15.
\displaystyle \text{Reason: Number of non-empty subsets of a set having }n\text{ elements are }2^n-1.
\displaystyle \text{(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.}
\displaystyle \text{(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.}
\displaystyle \text{(c) Assertion is correct, reason is incorrect.}
\displaystyle \text{(d) Assertion is incorrect, reason is correct.}
\displaystyle \text{Answer:}
\displaystyle \text{The set }\{a,b,c,d\}\text{ has }4\text{ elements.}
\displaystyle \text{Number of non-empty subsets}=2^n-1.
\displaystyle \therefore \text{Number of non-empty subsets}=2^4-1=15.
\displaystyle \text{Thus, the Assertion and the Reason are both correct.}
\displaystyle \text{The Reason correctly explains the Assertion.}
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 57: }\text{Suppose }A,\ B\text{ and }C\text{ are three arbitrary sets and }U\text{ is a universal set.}
\displaystyle \text{Assertion: If }B=U-A,\text{ then }n(B)=n(U)-n(A).
\displaystyle \text{Reason: If }C=A-B,\text{ then }n(C)=n(A)-n(B).
\displaystyle \text{(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.}
\displaystyle \text{(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.}
\displaystyle \text{(c) Assertion is correct, reason is incorrect.}
\displaystyle \text{(d) Assertion is incorrect, reason is correct.}
\displaystyle \text{Answer:}
\displaystyle \text{Since }B=U-A,\text{ the sets }A\text{ and }B\text{ are disjoint and }A\cup B=U.
\displaystyle \therefore n(U)=n(A)+n(B).
\displaystyle \therefore n(B)=n(U)-n(A).
\displaystyle \text{Hence, the Assertion is correct.}
\displaystyle \text{For }C=A-B,\text{ in general }n(C)=n(A)-n(A\cap B),\text{ not }n(A)-n(B).
\displaystyle \text{Hence, the Reason is incorrect.}
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 58: }\text{Assertion: Let }A=\{1,\{2,3\}\},\text{ then }P(A)=\{\{1\},\{2,3\},\emptyset,\{1,\{2,3\}\}\}.
\displaystyle \text{Reason: Power set is set of all subsets of }A.
\displaystyle \text{(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.}
\displaystyle \text{(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.}
\displaystyle \text{(c) Assertion is correct, reason is incorrect.}
\displaystyle \text{(d) Assertion is incorrect, reason is correct.}
\displaystyle \text{Answer:}
\displaystyle A=\{1,\{2,3\}\}\text{ has two elements, namely }1\text{ and }\{2,3\}.
\displaystyle \text{Therefore, its subsets are }\emptyset,\ \{1\},\ \{\{2,3\}\}\text{ and }\{1,\{2,3\}\}.
\displaystyle \therefore P(A)=\{\emptyset,\{1\},\{\{2,3\}\},\{1,\{2,3\}\}\}.
\displaystyle \text{The Assertion incorrectly uses }\{2,3\}\text{ instead of }\{\{2,3\}\}\text{ as an element of }P(A).
\displaystyle \text{Hence, the Assertion is incorrect.}
\displaystyle \text{The Reason is correct because a power set is the set of all subsets of a given set.}
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 59: }\text{Assertion: The subsets of the set }\{1,\{2\}\}\text{ are }\{\},\{1\},\{\{2\}\}\text{ and }\{1,\{2\}\}.
\displaystyle \text{Reason: The total number of proper subsets of a set containing }n\text{ elements is } \\ 2^n-1.
\displaystyle \text{(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.}
\displaystyle \text{(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.}
\displaystyle \text{(c) Assertion is correct, reason is incorrect.}
\displaystyle \text{(d) Assertion is incorrect, reason is correct.}
\displaystyle \text{Answer:}
\displaystyle \text{The set }\{1,\{2\}\}\text{ has two elements, namely }1\text{ and }\{2\}.
\displaystyle \text{Therefore, its subsets are }\emptyset,\{1\},\{\{2\}\}\text{ and }\{1,\{2\}\}.
\displaystyle \text{Hence, the Assertion is correct.}
\displaystyle \text{A set having }n\text{ elements has }2^n\text{ subsets.}
\displaystyle \text{Therefore, the number of proper subsets is }2^n-1.
\displaystyle \text{Hence, the Reason is correct, but it does not explain which subsets the given set has.}
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 60: }\text{Assertion: For any two sets }A\text{ and }B,\ A-B\subset B'.
\displaystyle \text{Reason: If }A\text{ be any set, then }A\cap A'=\emptyset.
\displaystyle \text{(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.}
\displaystyle \text{(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.}
\displaystyle \text{(c) Assertion is correct, reason is incorrect.}
\displaystyle \text{(d) Assertion is incorrect, reason is correct.}
\displaystyle \text{Answer:}
\displaystyle A-B=A\cap B'.
\displaystyle \text{Since }A\cap B'\subset B',\text{ we have }A-B\subset B'.
\displaystyle \text{Hence, the Assertion is correct.}
\displaystyle \text{Also, a set and its complement have no common element.}
\displaystyle \therefore A\cap A'=\emptyset.
\displaystyle \text{Hence, the Reason is correct.}
\displaystyle \text{However, the Reason does not directly explain why }A-B\subset B'.
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

 

\displaystyle \text{INTEGER TYPE QUESTIONS}

\displaystyle \text{Directions : This section contains integer type questions. The answer to each of the question}
\displaystyle \text{is a single digit integer, ranging from 0 to 9. Choose the correct option.}

\displaystyle \textbf{Question 61: }\text{If }X=\{1,2,3,\ldots,10\}\text{ and `}a\text{' represents any element of }X,\text{ then the set}
\displaystyle \text{containing all the elements satisfy }a+2=6,\ a\in X\text{ is:}
\displaystyle \text{(a) }\{4\}\qquad\text{(b) }\{3\}\qquad\text{(c) }\{2\}\qquad\text{(d) }\{5\}
\displaystyle \text{Answer:}
\displaystyle \text{Given, }a+2=6.
\displaystyle \therefore a=6-2=4.
\displaystyle \text{Since }4\in X,\text{ the required set is }\{4\}.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 62: }\text{If a set is denoted as }B=\emptyset,\text{ then the number of element in }B\text{ is:}
\displaystyle \text{(a) }3\qquad\text{(b) }2\qquad\text{(c) }1\qquad\text{(d) }0
\displaystyle \text{Answer:}
\displaystyle B=\emptyset\text{ represents the empty set, which contains no element.}
\displaystyle \therefore n(B)=0.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 63: }\text{The number of non-empty subsets of the set }\{1,2,3,4\}\text{ is }3a.\text{ The value of }
\displaystyle a\text{ is:}
\displaystyle \text{(a) }3\qquad\text{(b) }4\qquad\text{(c) }5\qquad\text{(d) }6
\displaystyle \text{Answer:}
\displaystyle \text{The set }\{1,2,3,4\}\text{ contains }4\text{ elements.}
\displaystyle \text{Number of non-empty subsets}=2^4-1=16-1=15.
\displaystyle \text{According to the question, }3a=15.
\displaystyle \therefore a=5.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 64: }\text{If }X=\{1,2,3\},\text{ then the number of proper subsets is:}
\displaystyle \text{(a) }5\qquad\text{(b) }6\qquad\text{(c) }7\qquad\text{(d) }8
\displaystyle \text{Answer:}
\displaystyle \text{The set }X\text{ contains }3\text{ elements.}
\displaystyle \text{Therefore, the total number of subsets}=2^3=8.
\displaystyle \text{A proper subset cannot be equal to the set itself.}
\displaystyle \therefore \text{Number of proper subsets}=8-1=7.
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

\displaystyle \textbf{Question 65: }\text{If }A=\{(x,y):x^2+y^2=25\}\text{ and }
\displaystyle B=\{(x,y):x^2+9y^2=144\}, \ \text{then the number of points }A\cap B\text{ contains is:}
\displaystyle \text{(a) }1\qquad\text{(b) }2\qquad\text{(c) }3\qquad\text{(d) }4
\displaystyle \text{Answer:}
\displaystyle \text{For a point to belong to }A\cap B,\text{ it must satisfy both equations.}
\displaystyle x^2+y^2=25\qquad\text{and}\qquad x^2+9y^2=144.
\displaystyle \text{Subtracting the first equation from the second,}
\displaystyle 8y^2=119.
\displaystyle \therefore y^2=\frac{119}{8}.
\displaystyle \text{Substituting in }x^2+y^2=25,
\displaystyle x^2=25-\frac{119}{8}=\frac{81}{8}.
\displaystyle \text{Thus, }x\text{ has two possible values and }y\text{ also has two possible values.}
\displaystyle \therefore \text{There are }2\times2=4\text{ points in }A\cap B.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

 

\displaystyle \text{CASE BASED QUESTIONS}

\displaystyle \textbf{Case Study - 1}

\displaystyle \text{Venn diagrams were invented by a logician John Venn as a way of picturing relationships}
\displaystyle \text{between different groups of things. These diagrams, also called Set diagrams or Logic diagrams,}
\displaystyle \text{are widely used in mathematics, statistics, logic, teaching, linguistics, computer science and}
\displaystyle \text{business.}
\displaystyle \text{In the following diagram, triangle shows children, circle shows rural population, rectangle shows}
\displaystyle \text{school going population \& square shows boys.}

\displaystyle \text{Based on the information stated above, answer the below given questions:}

\displaystyle \text{Case Study:}
\displaystyle \text{Let }C=\text{Children},\quad R=\text{Rural population},\quad S=\text{School-going population},
\displaystyle \text{and }B=\text{Boys}.
\displaystyle \text{From the given Venn diagram, we identify the numbered regions according to their positions.}
\displaystyle \\

\displaystyle \textbf{Question 66: }\text{(i) The rural boys not going to school are denoted by which}
\displaystyle \text{number?}
\displaystyle \text{(a) }1\qquad\text{(b) }2\qquad\text{(c) }1,2\qquad\text{(d) }2,8
\displaystyle \text{Answer:}
\displaystyle \text{We require the region representing boys who belong to the rural population but are}
\displaystyle \text{not school-going.}
\displaystyle \text{From the diagram, this region is denoted by }2.
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 67: }\text{(ii) The children from rural population not going to school are}
\displaystyle \text{denoted by which number?}
\displaystyle \text{(a) }1\qquad\text{(b) }2\qquad\text{(c) }6\qquad\text{(d) }2,6
\displaystyle \text{Answer:}
\displaystyle \text{We require children who belong to the rural population but are outside the school-going}
\displaystyle \text{region.}
\displaystyle \text{From the diagram, these regions are denoted by }2\text{ and }6.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 68: }\text{(iii) What is represented by number }4\text{?}
\displaystyle \text{(a) Children who are not from rural population}\qquad\text{(b) Children who are boys}
\displaystyle \text{(c) School going boys}\qquad\text{(d) School going boys who are not from rural}
\displaystyle \text{population}
\displaystyle \text{Answer:}
\displaystyle \text{Region }4\text{ lies in the children, boys and school-going regions, but outside the rural}
\displaystyle \text{population.}
\displaystyle \therefore 4\text{ represents school going boys who are not from rural population.}
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 69: }\text{(iv) School going boys from village are denoted by which number?}
\displaystyle \text{(a) }3\qquad\text{(b) }3,5\qquad\text{(c) }3,4\qquad\text{(d) }3,4,5,7
\displaystyle \text{Answer:}
\displaystyle \text{Here, village refers to the rural population. We require the common region of school-going,}
\displaystyle \text{boys and rural population.}
\displaystyle \text{From the diagram, the corresponding regions are }3\text{ and }5.
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 70: }\text{(v) Number of children who are not from rural population?}
\displaystyle \text{(a) }8\qquad\text{(b) }2\qquad\text{(c) }7\qquad\text{(d) }9
\displaystyle \text{Answer:}
\displaystyle \text{From the given options and the intended classification in the diagram, the region}
\displaystyle \text{representing children who are not from the rural population is denoted by }9.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

 

\displaystyle \textbf{Case Study - 2}

\displaystyle \text{In D.A.V School, Bahadurgarh, a survey was done on }400\text{ students. It was found that }100
\displaystyle \text{like to take apple juice, }150\text{ like to take orange juice and }75\text{ like both apple as well as orange}
\displaystyle \text{juice.} \displaystyle \text{Based on above information, answer the following questions:}

\displaystyle \text{Case Study:}
\displaystyle \text{In D.A.V School, Bahadurgarh, a survey was done on }400\text{ students. It was found that }100
\displaystyle \text{like to take apple juice, }150\text{ like to take orange juice and }75\text{ like both apple as well as orange}
\displaystyle \text{juice. Based on the above information, answer the following questions:}
\displaystyle \text{Let }A=\text{set of students who like apple juice and }B=\text{set of students who like orange juice.}
\displaystyle n(A)=100,\qquad n(B)=150,\qquad n(A\cap B)=75,\qquad n(U)=400.
\displaystyle \\

\displaystyle \textbf{Question 71: }\text{(i) Number of students who like either of the drink:}
\displaystyle \text{(a) }400\qquad\text{(b) }175\qquad\text{(c) }250\qquad\text{(d) }325
\displaystyle \text{Answer:}
\displaystyle \text{Number of students who like at least one of the two drinks}=n(A\cup B).
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B).
\displaystyle =100+150-75=175.
\displaystyle \therefore \text{The correct option is (b).}
\displaystyle \\

\displaystyle \textbf{Question 72: }\text{(ii) Number of students who likes neither apple juice nor orange juice:}
\displaystyle \text{(a) }225\qquad\text{(b) }325\qquad\text{(c) }75\qquad\text{(d) }25
\displaystyle \text{Answer:}
\displaystyle \text{Number of students who like neither drink}=n(U)-n(A\cup B).
\displaystyle =400-175=225.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 73: }\text{(iii) Number of students who likes only apple juice:}
\displaystyle \text{(a) }125\qquad\text{(b) }75\qquad\text{(c) }100\qquad\text{(d) }25
\displaystyle \text{Answer:}
\displaystyle \text{Number of students who like only apple juice}=n(A)-n(A\cap B).
\displaystyle =100-75=25.
\displaystyle \therefore \text{The correct option is (d).}
\displaystyle \\

\displaystyle \textbf{Question 74: }\text{(iv) Number of students who likes only orange juice:}
\displaystyle \text{(a) }75\qquad\text{(b) }25\qquad\text{(c) }100\qquad\text{(d) }125
\displaystyle \text{Answer:}
\displaystyle \text{Number of students who like only orange juice}=n(B)-n(A\cap B).
\displaystyle =150-75=75.
\displaystyle \therefore \text{The correct option is (a).}
\displaystyle \\

\displaystyle \textbf{Question 75: }\text{(v) Which information we get from the given data:}
\displaystyle \text{(a) }n(A\cup B)=n(A)+n(B)
\displaystyle \text{(b) }n(A\cup B)<n(A\cap B)
\displaystyle \text{(c) }n(A\cup B)<n(U)
\displaystyle \text{(d) }n(A\cap B)=n(A)+n(B)
\displaystyle \text{Answer:}
\displaystyle n(A\cup B)=175\text{ and }n(U)=400.
\displaystyle \therefore n(A\cup B)<n(U).
\displaystyle \text{Hence, the correct information obtained from the given data is }n(A\cup B)<n(U).
\displaystyle \therefore \text{The correct option is (c).}
\displaystyle \\

 

\displaystyle \text{1 MARK QUESTIONS}

\displaystyle \textbf{Question 76: }\text{Write down the set }\{5,25,125,625\}\text{ in the set builder form.}
\displaystyle \text{Answer:}
\displaystyle 5=5^1,\qquad 25=5^2,\qquad 125=5^3,\qquad 625=5^4.
\displaystyle \therefore \{5,25,125,625\}=\{x:x=5^n,\ n\in N,\ 1\leq n\leq4\}.
\displaystyle \\

\displaystyle \textbf{Question 77: }\text{If }n(A)=2\text{ then find }n[P(P(A))].
\displaystyle \text{Answer:}
\displaystyle n[P(A)]=2^{n(A)}=2^2=4.
\displaystyle \therefore n[P(P(A))]=2^{n[P(A)]}=2^4=16.
\displaystyle \\

\displaystyle \textbf{Question 78: }\text{Define equal sets with example.}
\displaystyle \text{Answer:}
\displaystyle \text{Two sets are said to be equal if they contain exactly the same elements.}
\displaystyle \text{For example, if }A=\{1,2,3\}\text{ and }B=\{3,2,1\},\text{ then }A=B.
\displaystyle \\

\displaystyle \textbf{Question 79: }\text{Write down }\{x:x\in R,\ -4<x\leq6\}\text{ as interval.}
\displaystyle \text{Answer:}
\displaystyle \text{Since }-4\text{ is not included and }6\text{ is included, the interval is }(-4,6].
\displaystyle \\

\displaystyle \textbf{Question 80: }\text{If }A=\{1,2,3,4,5,6\}\text{ and }B=\{2,4,6,8\},\text{ find }A-B.
\displaystyle \text{Answer:}
\displaystyle A-B\text{ consists of those elements which belong to }A\text{ but do not belong to }B.
\displaystyle A-B=\{1,3,5\}.
\displaystyle \\

\displaystyle \textbf{Question 81: }\text{Write the set }A=\{x:x\in R,\ 2x+11=15\}\text{ in the roster form.}
\displaystyle \text{Answer:}
\displaystyle 2x+11=15.
\displaystyle \therefore 2x=4.
\displaystyle \therefore x=2.
\displaystyle \text{Hence, in roster form, }A=\{2\}.
\displaystyle \\

\displaystyle \textbf{Question 82: }\text{Define disjoint sets with example.}
\displaystyle \text{Answer:}
\displaystyle \text{Two sets are said to be disjoint if they have no element in common.}
\displaystyle \text{Thus, }A\text{ and }B\text{ are disjoint if }A\cap B=\emptyset.
\displaystyle \text{For example, if }A=\{1,2,3\}\text{ and }B=\{4,5,6\},\text{ then }A\cap B=\emptyset.
\displaystyle \therefore A\text{ and }B\text{ are disjoint sets.}
\displaystyle \\

\displaystyle \textbf{Question 83: }\text{What universal set would you propose for the set of rectangles.}
\displaystyle \text{Answer:}
\displaystyle \text{Every rectangle is a quadrilateral.}
\displaystyle \therefore \text{The set of all quadrilaterals can be taken as the universal set.}
\displaystyle \\

\displaystyle \textbf{Question 84: }\text{If }A\text{ and }B\text{ are two sets such that }A\subset B,\text{ then what is }A\cup B\text{?}
\displaystyle \text{Answer:}
\displaystyle \text{Since }A\subset B,\text{ every element of }A\text{ is also an element of }B.
\displaystyle \therefore A\cup B=B.
\displaystyle \\

\displaystyle \textbf{Question 85: }\text{Write down the power set for the set }\{a,b\}.
\displaystyle \text{Answer:}
\displaystyle \text{The subsets of }\{a,b\}\text{ are }\emptyset,\{a\},\{b\}\text{ and }\{a,b\}.
\displaystyle \therefore P(\{a,b\})=\{\emptyset,\{a\},\{b\},\{a,b\}\}.
\displaystyle \\

\displaystyle \textbf{Question 86: }\text{If }S=\{x:x\text{ is a positive multiple of }3\text{ less than }100\}
\displaystyle \text{ and }P=\{x:x\text{ is a} \ \text{prime number less than }20\}.\text{ Then find }n(S)+n(P).
\displaystyle \text{Answer:}
\displaystyle S=\{3,6,9,\ldots,99\}.
\displaystyle \therefore n(S)=\frac{99}{3}=33.
\displaystyle P=\{2,3,5,7,11,13,17,19\}.
\displaystyle \therefore n(P)=8.
\displaystyle \therefore n(S)+n(P)=33+8=41.
\displaystyle \\

\displaystyle \textbf{Question 87: }\text{Two finite sets have }m\text{ and }n\text{ elements. The total number of}
\displaystyle \text{subsets of the first set is }56\text{ more than the total number of subsets of the second}
\displaystyle \text{set. Find the values of }m\text{ and }n.
\displaystyle \text{Answer:}
\displaystyle \text{A set having }m\text{ elements has }2^m\text{ subsets, while a set having }n\text{ elements has }2^n\text{ subsets.}
\displaystyle \text{According to the given condition,}
\displaystyle 2^m-2^n=56.
\displaystyle \text{Now, }56=64-8=2^6-2^3.
\displaystyle \therefore m=6\text{ and }n=3.
\displaystyle \\

\displaystyle \textbf{Question 88: }\text{Let }S=\text{ the set of all triangles, }P=\text{ the set of all}
\displaystyle \text{isosceles triangles, }Q=\text{ the set of all equilateral triangles, }R=\text{ the set of all}
\displaystyle \text{right-angled triangles. What do the sets }P\cap Q\text{ and }R-P\text{ represent}
\displaystyle \text{respectively?}
\displaystyle \text{Answer:}
\displaystyle \text{Every equilateral triangle is also an isosceles triangle. Hence, }Q\subset P.
\displaystyle \therefore P\cap Q=Q.
\displaystyle \text{Thus, }P\cap Q\text{ represents the set of all equilateral triangles.}
\displaystyle R-P\text{ represents the set of right-angled triangles which are not isosceles triangles.}
\displaystyle \therefore P\cap Q\text{ represents equilateral triangles and }R-P\text{ represents right-angled}
\displaystyle \text{triangles which are not isosceles.}
\displaystyle \\

\displaystyle \textbf{Question 89: }\text{Let }V=\{a,e,i,o,u\},\ V-B=\{e,o\}\text{ and }B-V=\{k\}.\text{ Find the set }B.
\displaystyle \text{Answer:}
\displaystyle V-B=\{e,o\}\text{ means that }e\text{ and }o\text{ are the elements of }V\text{ which are not in }B.
\displaystyle \therefore a,i,u\text{ belong to }B.
\displaystyle B-V=\{k\}\text{ means that }k\text{ is the element of }B\text{ which is not in }V.
\displaystyle \therefore B=\{a,i,k,u\}.
\displaystyle \\

\displaystyle \textbf{Question 90: }\text{If }A=\{a,\{b\}\},\text{ then find }P(A).
\displaystyle \text{Answer:}
\displaystyle A=\{a,\{b\}\}\text{ has two elements, namely }a\text{ and }\{b\}.
\displaystyle \text{Therefore, its subsets are }\emptyset,\{a\},\{\{b\}\}\text{ and }\{a,\{b\}\}.
\displaystyle \therefore P(A)=\{\emptyset,\{a\},\{\{b\}\},\{a,\{b\}\}\}.
\displaystyle \\

 

\displaystyle \text{2 MARKS QUESTIONS}

\displaystyle \textbf{Question 91: }\text{Write down all the subsets of the set }\{1,2,3\}.
\displaystyle \text{Answer:}
\displaystyle \text{The set }\{1,2,3\}\text{ has }3\text{ elements, so it has }2^3=8\text{ subsets.}
\displaystyle \therefore P(\{1,2,3\})=\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}.
\displaystyle \\

\displaystyle \textbf{Question 92: }\text{Draw appropriate Venn diagram for each of the following:}
\displaystyle \text{(a) }(A\cup B)'\qquad\qquad\text{(b) }A-B
\displaystyle \text{Answer:}
\displaystyle \text{(a) }(A\cup B)'\text{ represents all elements which are outside both }A\text{ and }B.
\displaystyle \therefore (A\cup B)'=A'\cap B'.
\displaystyle \text{In the Venn diagram, shade the region outside both circles }A\text{ and }B.
\displaystyle \text{(b) }A-B\text{ represents all elements which belong to }A\text{ but do not belong to }B.
\displaystyle \therefore A-B=A\cap B'.
\displaystyle \text{In the Venn diagram, shade the part of }A\text{ excluding }A\cap B.
\displaystyle \\

\displaystyle \textbf{Question 93: }\text{Write the set }A=\{5,25,125,625\}\text{ in set-builder form.}
\displaystyle \text{Answer:}
\displaystyle 5=5^1,\qquad25=5^2,\qquad125=5^3,\qquad625=5^4.
\displaystyle \therefore A=\{x:x=5^n,\ n\in N,\ 1\leq n\leq4\}.
\displaystyle \\

\displaystyle \textbf{Question 94: }\text{Prove that }A=B\text{ if }P(A)=P(B).
\displaystyle \text{Answer:}
\displaystyle \text{Given }P(A)=P(B).
\displaystyle \text{Since }A\in P(A),\text{ we have }A\in P(B).
\displaystyle \therefore A\subseteq B.
\displaystyle \text{Similarly, since }B\in P(B)=P(A),\text{ we have }B\subseteq A.
\displaystyle \therefore A=B.
\displaystyle \\

\displaystyle \textbf{Question 95: }\text{A survey shows that }63\%\text{ of Indians like cheese whereas }76\%\text{ like apples.}
\displaystyle \text{If }x\%\text{ of Indians like cheese and apples, find the value of }x.
\displaystyle \text{Answer:}
\displaystyle \text{Let }C\text{ and }A\text{ denote the sets of Indians who like cheese and apples respectively.}
\displaystyle n(C)=63,\qquad n(A)=76,\qquad n(C\cap A)=x.
\displaystyle n(C\cup A)=63+76-x=139-x.
\displaystyle \text{Since }n(C\cup A)\leq100,\text{ we get }139-x\leq100.
\displaystyle \therefore x\geq39.
\displaystyle \text{Also, }x\leq\min(63,76)=63.
\displaystyle \therefore 39\leq x\leq63.
\displaystyle \text{Hence, the exact value of }x\text{ cannot be determined; }39\leq x\leq63.
\displaystyle \\

\displaystyle \textbf{Question 96: }\text{If }n(A)=4,\text{ how many proper subsets does the set }A\text{ have?}
\displaystyle \text{Answer:}
\displaystyle \text{A set having }4\text{ elements has }2^4=16\text{ subsets.}
\displaystyle \text{A proper subset is any subset other than the set itself.}
\displaystyle \therefore \text{Number of proper subsets}=16-1=15.
\displaystyle \\

\displaystyle \textbf{Question 97: }\text{Using properties of sets prove that }(A\cup B)'=A'\cap B'.
\displaystyle \text{Answer:}
\displaystyle \text{Let }x\in(A\cup B)'.
\displaystyle \therefore x\notin A\cup B.
\displaystyle \therefore x\notin A\text{ and }x\notin B.
\displaystyle \therefore x\in A'\text{ and }x\in B'.
\displaystyle \therefore x\in A'\cap B'.
\displaystyle \text{Hence, }(A\cup B)'\subseteq A'\cap B'.
\displaystyle \text{Conversely, let }x\in A'\cap B'.
\displaystyle \therefore x\notin A\text{ and }x\notin B\Rightarrow x\notin A\cup B.
\displaystyle \therefore x\in(A\cup B)'.
\displaystyle \text{Hence, }A'\cap B'\subseteq(A\cup B)'.
\displaystyle \therefore (A\cup B)'=A'\cap B'.
\displaystyle \\

\displaystyle \textbf{Question 98: }\text{Using Venn Diagram prove that } \\ A\cap(B-C)=(A\cap B)-(A\cap C).
\displaystyle \text{Answer:}
\displaystyle B-C=B\cap C'.
\displaystyle \therefore A\cap(B-C)=A\cap B\cap C'.
\displaystyle \text{Thus, in the Venn diagram, shade the region common to }A\text{ and }B\text{ but outside }C.
\displaystyle \text{Now, }(A\cap B)-(A\cap C)=(A\cap B)\cap(A\cap C)'.
\displaystyle =(A\cap B)\cap(A'\cup C').
\displaystyle =(A\cap B\cap A')\cup(A\cap B\cap C').
\displaystyle =\emptyset\cup(A\cap B\cap C').
\displaystyle =A\cap B\cap C'.
\displaystyle \text{Both sides represent the same shaded region in the Venn diagram.}
\displaystyle \therefore A\cap(B-C)=(A\cap B)-(A\cap C).
\displaystyle \\

\displaystyle \textbf{Question 99: }\text{For all sets }A,\ B\text{ and }C,\text{ if }A\subset B,\text{ then }A\cup C\subset B\cup C.
\displaystyle \text{Answer:}
\displaystyle \text{Given }A\subset B.
\displaystyle \text{Let }x\in A\cup C.
\displaystyle \therefore x\in A\text{ or }x\in C.
\displaystyle \text{If }x\in A,\text{ then }x\in B,\text{ since }A\subset B.
\displaystyle \text{Thus, in either case, }x\in B\text{ or }x\in C.
\displaystyle \therefore x\in B\cup C.
\displaystyle \therefore A\cup C\subset B\cup C.
\displaystyle \\

\displaystyle \textbf{Question 100: }\text{For all sets }A\text{ and }B,\ (A-B)\cup(A\cap B)=A.
\displaystyle \text{Answer:}
\displaystyle A-B=A\cap B'.
\displaystyle \therefore (A-B)\cup(A\cap B)=(A\cap B')\cup(A\cap B).
\displaystyle =A\cap(B'\cup B).
\displaystyle =A\cap U.
\displaystyle =A.
\displaystyle \therefore (A-B)\cup(A\cap B)=A.
\displaystyle \\

 

\displaystyle \text{4 MARKS QUESTIONS}

\displaystyle \textbf{Question 101: }\text{If }A=\{3,5,7,9,11\},\ B=\{7,9,11,13\},\ C=\{11,13,15\},
\displaystyle D=\{15,17\},\text{ find:}
\displaystyle \text{(a) }(A\cup B)\cap C\qquad\text{(b) }C-D\qquad\text{(c) }(A\cap B)\cap(B\cup C)\qquad\text{(d) }B\cap D
\displaystyle \text{Answer:}
\displaystyle \text{(a) }A\cup B=\{3,5,7,9,11,13\}.
\displaystyle \therefore (A\cup B)\cap C=\{3,5,7,9,11,13\}\cap\{11,13,15\}=\{11,13\}.
\displaystyle \text{(b) }C-D=\{11,13,15\}-\{15,17\}=\{11,13\}.
\displaystyle \text{(c) }A\cap B=\{7,9,11\}.
\displaystyle B\cup C=\{7,9,11,13,15\}.
\displaystyle \therefore (A\cap B)\cap(B\cup C)=\{7,9,11\}.
\displaystyle \text{(d) }B\cap D=\{7,9,11,13\}\cap\{15,17\}=\emptyset.
\displaystyle \\

\displaystyle \textbf{Question 102: }\text{If }S\text{ and }T\text{ are two sets such that }S\text{ has }21\text{ elements, }T\text{ has }32
\displaystyle \text{elements, and }S\cap T\text{ has }11\text{ elements, how many elements does }S\cup T\text{ have?}
\displaystyle \text{Answer:}
\displaystyle n(S)=21,\qquad n(T)=32,\qquad n(S\cap T)=11.
\displaystyle \text{Using the addition principle for two sets,}
\displaystyle n(S\cup T)=n(S)+n(T)-n(S\cap T).
\displaystyle =21+32-11.
\displaystyle =42.
\displaystyle \therefore S\cup T\text{ has }42\text{ elements.}
\displaystyle \\

\displaystyle \textbf{Question 103: }\text{In a survey of }100\text{ students the number of students studying the various}
\displaystyle \text{languages were found to be: English only }18,\text{ English but not Hindi }23,\text{ English and}
\displaystyle \text{Sanskrit }8,\text{ English }26,\text{ Sanskrit }48,\text{ Sanskrit and Hindi }8,\text{ no language }24.\text{ Find:}
\displaystyle \text{(i) How many students were studying Hindi?}
\displaystyle \text{(ii) How many students were studying English and Hindi?}
\displaystyle \text{(iii) How many students were studying Sanskrit only?}
\displaystyle \text{Answer:}
\displaystyle \text{Let }E,\ H\text{ and }S\text{ denote the sets of students studying English, Hindi and Sanskrit respectively.}
\displaystyle n(E)=26,\qquad n(S)=48,\qquad n(E\text{ only})=18.
\displaystyle n(E\cap S)=8,\qquad n(S\cap H)=8.
\displaystyle \text{Since }23\text{ study English but not Hindi,}
\displaystyle n(E\cap S\text{ only})=23-18=5.
\displaystyle \therefore n(E\cap H\cap S)=8-5=3.
\displaystyle \text{Now, }n(E)=18+5+n(E\cap H\text{ only})+3=26.
\displaystyle \therefore n(E\cap H\text{ only})=0.
\displaystyle \text{Also, }n(S\cap H\text{ only})=8-3=5.
\displaystyle \text{Let the number studying Sanskrit only be }x.
\displaystyle 48=x+5+5+3.
\displaystyle \therefore x=35.
\displaystyle \text{Thus, the number studying Sanskrit only is }35.
\displaystyle \text{Since }24\text{ students study no language,}
\displaystyle n(E\cup H\cup S)=100-24=76.
\displaystyle \text{Let the number studying Hindi only be }y.
\displaystyle 76=18+5+0+3+5+35+y.
\displaystyle \therefore y=10.
\displaystyle \text{(i) }n(H)=10+0+5+3=18.
\displaystyle \therefore \text{Number of students studying Hindi}=18.
\displaystyle \text{(ii) }n(E\cap H)=0+3=3.
\displaystyle \therefore \text{Number of students studying English and Hindi}=3.
\displaystyle \text{(iii) Number of students studying Sanskrit only}=35.
\displaystyle \\

\displaystyle \textbf{Question 104: }\text{Define the following with the help of an example:}
\displaystyle \text{(a) Null Set}\qquad\text{(b) Singleton Set}\qquad\text{(c) Sub-Set}\qquad\text{(d) Power Set.}
\displaystyle \text{Answer:}
\displaystyle \text{(a) Null Set: A set which contains no element is called a null or empty set.}
\displaystyle \text{For example, }\{x:x\in N,\ 2<x<3\}=\emptyset.
\displaystyle \text{(b) Singleton Set: A set containing exactly one element is called a singleton set.}
\displaystyle \text{For example, }\{x:x\in N,\ x^2=4\}=\{2\}.
\displaystyle \text{(c) Sub-Set: A set }A\text{ is a subset of }B\text{ if every element of }A\text{ is also an element of }B.
\displaystyle \text{For example, if }A=\{1,2\}\text{ and }B=\{1,2,3\},\text{ then }A\subset B.
\displaystyle \text{(d) Power Set: The set of all subsets of a set }A\text{ is called the power set of }A.
\displaystyle \text{For example, if }A=\{1,2\},\text{ then }P(A)=\{\emptyset,\{1\},\{2\},\{1,2\}\}.
\displaystyle \\

\displaystyle \textbf{Question 105: }\text{In each of the following, determine whether the given statement is}
\displaystyle \text{true or false. If true prove it, and if false, give an example.}
\displaystyle \text{(a) If }x\in A\text{ and }A\in B,\text{ then }x\in B
\displaystyle \text{(b) If }x\in A\text{ and }A\subset B,\text{ then }x\in B
\displaystyle \text{Answer:}

\displaystyle \text{(a) The given statement is false.}
\displaystyle \text{For example, let }A=\{1\}\text{ and }B=\{\{1\}\}.
\displaystyle \text{Take }x=1.
\displaystyle \text{Then }x\in A\text{ and }A=\{1\}\in B.
\displaystyle \text{However, }1\notin B,\text{ since the only element of }B\text{ is the set }\{1\}.
\displaystyle \therefore x\in A\text{ and }A\in B\text{ do not necessarily imply }x\in B.
\displaystyle \text{Hence, the statement is false.}

\displaystyle \text{(b) The given statement is true.}
\displaystyle \text{Given }x\in A\text{ and }A\subset B.
\displaystyle \text{By the definition of a subset, every element of }A\text{ is also an element of }B.
\displaystyle \therefore x\in B.
\displaystyle \text{Hence, the statement is true.}
\displaystyle \\

\displaystyle \textbf{Question 106: }\text{In a survey of }25\text{ students of a school it was found that }15
\displaystyle \text{study Mathematics, }12\text{ study Physics, }11\text{ study Chemistry, }9\text{ study both}
\displaystyle \text{Mathematics \& Physics, }4\text{ study both Physics \& Chemistry, }5\text{ study both}
\displaystyle \text{Chemistry and Mathematics and }3\text{ students all of the above three subjects. Find the}
\displaystyle \text{number of students who study:}
\displaystyle \text{(a) Only Mathematics}
\displaystyle \text{(b) At least one of the three subjects.}
\displaystyle \text{Answer:}

\displaystyle \text{Let }M,\ P\text{ and }C\text{ denote the sets of students studying Mathematics, Physics and Chemistry.}
\displaystyle n(M)=15,\qquad n(P)=12,\qquad n(C)=11.
\displaystyle n(M\cap P)=9,\qquad n(P\cap C)=4,\qquad n(C\cap M)=5.
\displaystyle n(M\cap P\cap C)=3.

\displaystyle \text{(a) Number of students studying only Mathematics is}
\displaystyle n(M)-n(M\cap P)-n(M\cap C)+n(M\cap P\cap C).
\displaystyle =15-9-5+3.
\displaystyle =4.
\displaystyle \therefore \text{Number of students studying only Mathematics}=4.

\displaystyle \text{(b) Students studying at least one subject are represented by }M\cup P\cup C.
\displaystyle \text{By the inclusion-exclusion principle,}
\displaystyle n(M\cup P\cup C)=n(M)+n(P)+n(C)
\displaystyle \qquad-n(M\cap P)-n(P\cap C)-n(C\cap M)+n(M\cap P\cap C).
\displaystyle =15+12+11-9-4-5+3.
\displaystyle =23.
\displaystyle \therefore \text{Number of students studying at least one of the three subjects}=23.
\displaystyle \\

\displaystyle \textbf{Question 107: }\text{In a group of students, }100\text{ students know Hindi, }50\text{ know}
\displaystyle \text{English and }25\text{ know both. Each of the students know either Hindi or English. How}
\displaystyle \text{many students are there in the group?}
\displaystyle \text{Answer:}
\displaystyle \text{Let }H\text{ be the set of students who know Hindi and }E\text{ be the set of students who know English.}
\displaystyle n(H)=100,\qquad n(E)=50,\qquad n(H\cap E)=25.
\displaystyle \text{Since every student knows either Hindi or English, the total number of students is }n(H\cup E).
\displaystyle n(H\cup E)=n(H)+n(E)-n(H\cap E).
\displaystyle =100+50-25=125.
\displaystyle \therefore \text{There are }125\text{ students in the group.}
\displaystyle \\

\displaystyle \textbf{Question 108: }\text{If }n(A-B)=18,\ n(A\cup B)=70\text{ and }n(A\cap B)=25,\text{ then find }n(B).
\displaystyle \text{Answer:}
\displaystyle A=(A-B)\cup(A\cap B),\text{ and these two sets are disjoint.}
\displaystyle \therefore n(A)=n(A-B)+n(A\cap B).
\displaystyle =18+25=43.
\displaystyle \text{Using }n(A\cup B)=n(A)+n(B)-n(A\cap B),
\displaystyle 70=43+n(B)-25.
\displaystyle \therefore 70=18+n(B).
\displaystyle \therefore n(B)=52.
\displaystyle \\

\displaystyle \textbf{Question 109: }\text{If }L=\{1,2,3,4\},\ M=\{3,4,5,6\}\text{ and }N=\{1,3,5\},\text{ then verify that}
\displaystyle L-(M\cup N)=(L-M)\cap(L-N).
\displaystyle \text{Answer:}
\displaystyle M\cup N=\{1,3,4,5,6\}.
\displaystyle \therefore L-(M\cup N)=\{1,2,3,4\}-\{1,3,4,5,6\}=\{2\}.
\displaystyle \text{Now, }L-M=\{1,2\}\text{ and }L-N=\{2,4\}.
\displaystyle \therefore (L-M)\cap(L-N)=\{1,2\}\cap\{2,4\}=\{2\}.
\displaystyle \therefore L-(M\cup N)=(L-M)\cap(L-N).
\displaystyle \text{Hence, the given identity is verified.}
\displaystyle \\

\displaystyle \textbf{Question 110: }\text{If }A,\ B\text{ and }C\text{ be sets. Then, show that}
\displaystyle A\cap(B\cup C)=(A\cap B)\cup(A\cap C).
\displaystyle \text{Answer:}
\displaystyle \text{Let }x\in A\cap(B\cup C).
\displaystyle \therefore x\in A\text{ and }x\in B\cup C.
\displaystyle \therefore x\in A\text{ and }(x\in B\text{ or }x\in C).
\displaystyle \therefore (x\in A\cap B)\text{ or }(x\in A\cap C).
\displaystyle \therefore x\in(A\cap B)\cup(A\cap C).
\displaystyle \therefore A\cap(B\cup C)\subseteq(A\cap B)\cup(A\cap C).
\displaystyle \text{Conversely, let }x\in(A\cap B)\cup(A\cap C).
\displaystyle \therefore (x\in A\cap B)\text{ or }(x\in A\cap C).
\displaystyle \therefore x\in A\text{ and }(x\in B\text{ or }x\in C).
\displaystyle \therefore x\in A\cap(B\cup C).
\displaystyle \therefore (A\cap B)\cup(A\cap C)\subseteq A\cap(B\cup C).
\displaystyle \therefore A\cap(B\cup C)=(A\cap B)\cup(A\cap C).
\displaystyle \text{Hence proved.}
\displaystyle \\

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