\displaystyle \textbf{Question 1: }\text{Which of the following collections of objects are sets?}

\displaystyle \text{(i) All the months in a year}
\displaystyle \text{Answer:}
\displaystyle \text{YES. It is a well-defined collection of distinct objects. There are }12\text{ months in a year.}
\displaystyle \text{So, it is a definite set of elements.}
\displaystyle A=\{\text{January, February, March, April, May, June, July, August, September, October,}
\displaystyle \text{November, December}\}

\displaystyle \text{(ii) All the rivers flowing in UP}
\displaystyle \text{Answer:}
\displaystyle \text{YES. It is a well-defined collection of distinct objects.}
\displaystyle A=\{\text{Ganga, Yamuna, }\ldots\}

\displaystyle \text{(iii) All the planets in our solar system}
\displaystyle \text{Answer:}
\displaystyle \text{YES. It is a well-defined collection of distinct objects.}
\displaystyle A=\{\text{Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}\}

\displaystyle \text{(iv) All the interesting dramas written by Premchand}
\displaystyle \text{Answer:}
\displaystyle \text{NO. It is not a well-defined collection of objects because the word ``interesting'' is}
\displaystyle \text{subjective. Had it been ``All dramas written by Premchand'', then it could have been a set}
\displaystyle \text{as that would be well defined.}

\displaystyle \text{(v) All the short boys in your class}
\displaystyle \text{Answer:}
\displaystyle \text{NO. It is not a well-defined collection of objects because the word ``short'' is subjective.}
\displaystyle \text{Had it been ``All boys shorter than }5\text{ feet }3\text{ inches'', then it would have become}
\displaystyle \text{well defined. We need a benchmark to have a well-defined list of objects.}

\displaystyle \text{(vi) All the letters of the English alphabet which precede K}
\displaystyle \text{Answer:}
\displaystyle \text{YES. It is a well-defined collection of distinct objects.}
\displaystyle A=\{A,B,C,D,E,F,G,H,I,J\}

\displaystyle \text{(vii) All the pet dogs in Nagpur}
\displaystyle \text{Answer:}
\displaystyle \text{YES. It is a well-defined collection of distinct objects. At any given time, whether a dog}
\displaystyle \text{is a pet and whether it is in Nagpur can be determined. The list may be large, but it is still}
\displaystyle \text{a well-defined collection of objects.}

\displaystyle \text{(viii) All the dishonest shop owners in Noida}
\displaystyle \text{Answer:}
\displaystyle \text{NO. It is not a well-defined collection because ``dishonest'' is subjective unless a specific}
\displaystyle \text{and objective criterion for dishonesty is given.}

\displaystyle \text{(ix) All the students in your school with age exceeding }15\text{ years}
\displaystyle \text{Answer:}
\displaystyle \text{YES. It is a well-defined collection of distinct objects. The school has a list of students}
\displaystyle \text{who study there. The set would include all those whose age is more than }15\text{ years.}

\displaystyle \text{(x) All the girls of Gita's class who are taller than Gita}
\displaystyle \text{Answer:}
\displaystyle \text{YES. It is a well-defined collection of distinct objects. We know all the girls studying in}
\displaystyle \text{the class and Gita's height can be measured. Hence the set would contain all the girls}
\displaystyle \text{in the class who are taller than Gita.}

\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Rewrite the following statements using the set notations:}

\displaystyle \text{(i) }p\text{ is an element of }A
\displaystyle \text{Answer:}
\displaystyle p\in A

\displaystyle \text{(ii) }q\text{ does not belong to set }B
\displaystyle \text{Answer:}
\displaystyle q\notin B

\displaystyle \text{(iii) }a\text{ and }b\text{ are members of set }C
\displaystyle \text{Answer:}
\displaystyle a,b\in C

\displaystyle \text{(iv) }B\text{ and }C\text{ are equivalent sets}
\displaystyle \text{Answer:}
\displaystyle n(B)=n(C)

\displaystyle \text{(v) Cardinal number of set }E\text{ is }15
\displaystyle \text{Answer:}
\displaystyle n(E)=15
\displaystyle \text{Note: The number of distinct elements contained in a finite set }A\text{ is called the cardinal}
\displaystyle \text{number of }A\text{ and is denoted by }n(A).
\displaystyle \text{For example, if }A=\{1,2,3,4,5\},\text{ then }n(A)=5.

\displaystyle \text{(vi) }A\text{ is an empty set and }B\text{ is a non-empty set}
\displaystyle \text{Answer:}
\displaystyle A=\phi\text{ and }B\neq\phi
\displaystyle \text{Note: A set consisting of no elements is called an empty set or a null set or a void set.}
\displaystyle \text{It is denoted by }\phi.\text{ We write }\phi=\{\}.

\displaystyle \text{(vii) }0\text{ is a whole number, but }0\text{ is not a natural number}
\displaystyle \text{Answer:}
\displaystyle 0\in W\text{ but }0\notin N
\displaystyle \text{Note: Whole numbers }(W):\text{ The numbers }\{0,1,2,3,\ldots\}.
\displaystyle \text{Natural numbers }(N):\text{ The counting numbers }\{1,2,3,\ldots\}.

\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Describe the following sets in roster form}

\displaystyle \text{(i) }B=\{x\mid x\in W,\ x\leq 6\}
\displaystyle \text{Answer:}
\displaystyle B=\{0,1,2,3,4,5,6\}

\displaystyle \text{(ii) }C=\{x\mid x\text{ is a factor of }32\}
\displaystyle \text{Answer:}
\displaystyle C=\{1,2,4,8,16,32\}
\displaystyle \text{Note: You can calculate the factors using a tree method.}

\displaystyle \text{(iii) }E=\{x\mid x=(2n+1),\ n\in W,\ n\leq 4\}
\displaystyle \text{Answer:}
\displaystyle E=\{1,3,5,7,9\}
\displaystyle \text{Note: }n\in W\text{ and }n\leq 4.\text{ This means }n\text{ is }0,1,2,3,\text{ and }4.
\displaystyle \text{Now calculate }x\text{ based on the given formula. Example: When }n=0,\ x=1.

\displaystyle \text{(iv) }F=\{x\mid x=n^2,\ n\in N,\ 2\leq n\leq 5\}
\displaystyle \text{Answer:}
\displaystyle F=\{4,9,16,25\}
\displaystyle \text{Note: }n\in N\text{ and }2\leq n\leq 5.\text{ This means }n\text{ is }2,3,4,5.
\displaystyle \text{Now substitute the value of }n\text{ to calculate }x.

\displaystyle \text{(v) }G=\left\{x\mid x=\frac{n}{n+3},\ n\in N\text{ and }n\leq 5\right\}
\displaystyle \text{Answer:}
\displaystyle G=\{1/4,2/5,3/6. \ 4/7,5/8\}
\displaystyle \text{Note: }n\in N\text{ and }n\leq 5,\text{ which means }n\text{ is }1,2,3,4,5.
\displaystyle \text{Now substitute the value of }n\text{ in the formula given for }x.

\displaystyle \text{(vi) }H=\{x\mid x\text{ is a two digit number, the sum of the digits is 8}\}
\displaystyle \text{Answer:}
\displaystyle H=\{17,26,35,44,53,62,71,80\}
\displaystyle \text{Note: If the first digit is }x,\text{ then the second digit is }(8-x).
\displaystyle \text{So now try the values of }x\text{ as }0,1,2,3,4,5,6,7,8,9,10\cdots
\displaystyle x\text{ cannot be }0\text{ as it is a two digit number. Also by same logic, }x\text{ cannot be}
\displaystyle \text{greater than }8\text{ either. Hence }x\text{ can only be }1,2,3,4,5,6,7,\text{ and }8\text{ only.}

\displaystyle \text{(vii) }I=\{x\mid x\in N,\ x\text{ is divisible by both 4 and 6 and }x\leq 60\}
\displaystyle \text{Answer:}
\displaystyle I=\{12,24,36,48,60\}
\displaystyle \text{Note: }x\leq 60\text{ means that }x=\{1,2,3\cdots 60\}.
\displaystyle \text{Of these numbers }x\text{ needs to be divisible both by }4\text{ and }6.

\displaystyle \text{(viii) }J=\{x\mid x=(1/n),\ n\in N\text{ and }n\leq 5\}
\displaystyle \text{Answer:}
\displaystyle J=\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5} \}
\displaystyle \text{Note: }n\in N\text{ and }n\leq 5\text{ which means that }n\text{ is }1,2,3,4,5.
\displaystyle \text{Now substitute the values.}

\displaystyle \text{(ix) }L=\{x\mid x\text{ is a letter of the word 'careless'}\}
\displaystyle \text{Answer:}
\displaystyle L=\{c,a,r,e,l,s\}
\displaystyle \text{Note: In careless, the alphabet }e\text{ and }s\text{ are repeated. We only need to take into}
\displaystyle \text{account distinct elements only.}

\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Describe the following sets in set builder form}

\displaystyle \text{(i) }A=\{5,6,7,8,9,10,11,12\}
\displaystyle \text{Answer:}
\displaystyle A=\{x\mid x\in N,\ 4<x<13\}\text{ or }A=\{x\mid x\in N,\ 5\leq x\leq 12\}

\displaystyle \text{(ii) }B=\{1,2,3,4,6,8,12,16,24,48\}
\displaystyle \text{Answer:}
\displaystyle B=\{x\mid x\text{ is a factor of }48\}

\displaystyle \text{(iii) }C=\{11,13,17,19,23,29,31,37\}
\displaystyle \text{Answer:}
\displaystyle C=\{x\mid x\text{ is a prime number, }x\in N\text{ and }10<x<40\}

\displaystyle \text{(iv) }D=\{21,23,25,27,29,31,33,35,37\}
\displaystyle \text{Answer:}
\displaystyle D=\{x\mid x\text{ is an odd number},\ 20<x<38\}\text{ or}
\displaystyle D=\{x\mid x=(2n+19),\ n\in N,\text{ and }1\leq n\leq 9\}

\displaystyle \text{(v) }I=\{9,16,25,36,49,64,81,100\}
\displaystyle \text{Answer:}
\displaystyle I=\{x\mid x=n^2,\ n\in N,\text{ and }3\leq n\leq 10\}
\displaystyle \text{Note: If you notice, the numbers are squares of }3,4,5,6,7,8,9,10.

\displaystyle \text{(vi) }J=\{-2,2\}
\displaystyle \text{Answer:}
\displaystyle J=\{x\mid x\in Z,\ x^2=4\}
\displaystyle \text{Note: Integers }(Z):\text{ Positive and negative counting numbers, as well as zero:}
\displaystyle \{\ldots,-2,-1,0,1,2,\ldots\}
\displaystyle x^2=4\text{ which means }x=\pm\sqrt{4},\text{ so }x\text{ can be }2\text{ or }-2.

\displaystyle \text{(vii) }K=\{0\}
\displaystyle \text{Answer:}
\displaystyle K=\{x\mid x=0\}

\displaystyle \text{(viii) }L=\{\}
\displaystyle \text{Answer:}
\displaystyle L=\{x\mid x\in N,\ x\neq x\}

\displaystyle \text{(ix) }M=\{a,b,c,d,e,f,g,h,i\}
\displaystyle \text{Answer:}
\displaystyle M=\{x\mid x\text{ is an English alphabet which precedes J}\}

\displaystyle \text{(x) }P=\{2/7,3/8,4/9,5/10,6/11,7/12,8/13,9/14\}
\displaystyle \text{Answer:}
\displaystyle P=\left\{x\mid x=\frac{n}{n+5},\ n\in N,\ 2\leq n\leq 9\right\}

\displaystyle \text{(xi) }S=\{\text{Atlantic, Arctic}\}
\displaystyle \text{Answer:}
\displaystyle S=\{x\mid x\text{ is an ocean name that starts with A}\}

\displaystyle \text{(xii) }T=\{\text{Mars, Mercury}\}
\displaystyle \text{Answer:}
\displaystyle T=\{x\mid x\text{ is a planet whose name starts with an M}\}

\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Separate finite and infinite sets from the following:}

\displaystyle \text{(i) Set of leaves on a tree}
\displaystyle \text{Answer:}
\displaystyle \text{Finite. This is because in this case, the process of counting the leaves would surely come}
\displaystyle \text{to an end.}

\displaystyle \text{(ii) Set of all counting numbers}
\displaystyle \text{Answer:}
\displaystyle \text{Infinite. This is because there is no end to the numbers.}

\displaystyle \text{(iii) }\{x\mid x\in N,\ x>1000\}
\displaystyle \text{Answer:}
\displaystyle \text{Infinite. This is because there is no end to the numbers since }x>1000.

\displaystyle \text{(iv) }\{x\mid x\in W,\ x<5000\}
\displaystyle \text{Answer:}
\displaystyle \text{Finite. }W\text{ is the set of whole numbers, which includes }0\text{ and all natural numbers.}
\displaystyle \text{Since }x\text{ is less than }5000,\text{ there are finite numbers to be counted.}

\displaystyle \text{(v) }\{x\mid x\in Z,\ x<4\}
\displaystyle \text{Answer:}
\displaystyle \text{Infinite. }Z\text{ is the set of integers. Integers can be negative numbers too. So in this case,}
\displaystyle \text{though }x\text{ is limited to less than }4\text{ on the positive scale, it can continue indefinitely}
\displaystyle \text{on the negative scale.}

\displaystyle \text{(vi) Set of all triangles in a plane}
\displaystyle \text{Answer:}
\displaystyle \text{Infinite. This is because there can be infinitely many triangles in a plane.}

\displaystyle \text{(vii) Set of all points on the circumference of a circle}
\displaystyle \text{Answer:}
\displaystyle \text{Infinite. This is because there are infinitely many points on the circumference of a circle.}

\displaystyle \text{(viii) }\{1,2,3,1,2,3,1,2,3,\cdots\}
\displaystyle \text{Answer:}
\displaystyle \text{Finite. In a set, repeated elements are counted only once. Therefore, this set is simply}
\displaystyle \{1,2,3\}\text{ and contains only three distinct elements.}

\displaystyle \text{(ix) }\{x\mid x\in Q,\ 2<x<3\}
\displaystyle \text{Answer:}
\displaystyle \text{Infinite.}
\displaystyle \text{Note: Rational numbers }(Q):\text{ Numbers that can be expressed as a ratio of an integer to a}
\displaystyle \text{non-zero integer. All integers are rational, but the converse is not true.}

\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Which of the following are empty sets?}

\displaystyle \text{(i) }A=\{x\mid x\in N,\ x+5=5\}
\displaystyle \text{Answer:}
\displaystyle \text{Empty Set}
\displaystyle \text{Note: Natural Numbers }N=\{1,2,3,4,5,\ldots\}.
\displaystyle \text{Since }x+5=5\text{ is given, it means }x=0.
\displaystyle \text{But }0\text{ does not belong to }N\text{ and hence the set is empty or null.}

\displaystyle \text{(ii) }B=\{x\mid x\in N,\ 2x+3=6\}
\displaystyle \text{Answer:}
\displaystyle \text{Empty Set}
\displaystyle \text{Note: Natural Numbers }N=\{1,2,3,4,5,\ldots\}.
\displaystyle \text{Since }2x+3=6\text{ is given, it means }x=\frac{3}{2}=1.5.
\displaystyle \text{But }1.5\text{ does not belong to }N\text{ and hence the set is empty or null.}

\displaystyle \text{(iii) }C=\{x\mid x\in W,\ x+2<5\}
\displaystyle \text{Answer:}
\displaystyle \text{Not an Empty Set}
\displaystyle \text{Note: Whole Numbers }W=\{0,1,2,3,4,5,\ldots\}.
\displaystyle \text{Since }x+2<5\text{ is given, it means }x<3.
\displaystyle \text{Therefore, }x\text{ can be }0,1,2\text{ and hence it is not an Empty Set.}

\displaystyle \text{(iv) }D=\{x\mid x\in N,\ 1<x\leq2\}
\displaystyle \text{Answer:}
\displaystyle \text{Not an Empty Set}
\displaystyle \text{Note: Natural Numbers }N=\{1,2,3,4,5,\ldots\}.
\displaystyle \text{Since }1<x\leq2,\text{ it means that }x\text{ can only be }2.
\displaystyle \text{Hence }D=\{2\}\text{ and is not an Empty Set.}

\displaystyle \text{(v) }E=\{x\mid x\in N,\ x^2+4=0\}
\displaystyle \text{Answer:}
\displaystyle \text{Empty Set}
\displaystyle \text{Note: Natural Numbers }N=\{1,2,3,4,5,\ldots\}.
\displaystyle x^2+4=0\text{ means that }x^2=-4,\text{ which is not possible for any natural number.}
\displaystyle \text{Hence }E\text{ is an Empty Set.}

\displaystyle \text{(vi) }F=\{x\mid x\text{ is a prime number},\ 90<x<96\}
\displaystyle \text{Answer:}
\displaystyle \text{Empty Set}
\displaystyle \text{Note: If }90<x<96,\text{ then }x\text{ can be }91,92,93,94,95.
\displaystyle 91=7\times13,\quad92=2\times46,\quad93=3\times31,\quad94=2\times47,\quad95=5\times19.
\displaystyle \text{Therefore, none of these numbers is prime and hence }F\text{ is an Empty Set.}

\displaystyle \text{(vii) }G=\{x\mid x\text{ is an even prime number}\}
\displaystyle \text{Answer:}
\displaystyle \text{Not an Empty Set}
\displaystyle \text{Note: }2\text{ is an even prime number. Hence }G=\{2\}.

\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Which of the following are pairs of equivalent sets?}

\displaystyle \text{Note: Two finite sets }A\text{ and }B\text{ are said to be equivalent if }n(A)=n(B),
\displaystyle \text{that is, they have the same number of elements. An equivalent set is simply a set with an}
\displaystyle \text{equal number of elements. The sets do not have to have the same exact elements, just the}
\displaystyle \text{same number of elements.}

\displaystyle \text{(i) }A=\{2,3,5,7\}\text{ and }B=\{x:x\text{ is a whole number, }x<3\}
\displaystyle \text{Answer:}
\displaystyle \text{No}
\displaystyle \text{Note: }A=\{2,3,5,7\}\text{ and }B=\{0,1,2\}.
\displaystyle \text{Therefore, }n(A)=4\text{ while }n(B)=3.
\displaystyle \text{Hence }A\text{ and }B\text{ are not equivalent sets.}

\displaystyle \text{(ii) }C=\{x:x+2=2\}\text{ and }D=\phi
\displaystyle \text{Answer:}
\displaystyle \text{No}
\displaystyle \text{Note: }C=\{0\}\text{ and }D=\{\}.
\displaystyle \text{Therefore, }n(C)=1\text{ while }n(D)=0.
\displaystyle \text{Hence }C\text{ and }D\text{ are not equivalent sets.}

\displaystyle \text{(iii) }E=\{x:x\text{ is a natural number, }x<4\}\text{ and}
\displaystyle F=\{x:x\text{ is a whole number, }x<3\}
\displaystyle \text{Answer:}
\displaystyle \text{Yes}
\displaystyle \text{Note: }E=\{1,2,3\}\text{ and }F=\{0,1,2\}.
\displaystyle \text{Therefore, }n(E)=3\text{ and }n(F)=3.
\displaystyle \text{Hence }E\text{ and }F\text{ are equivalent sets.}

\displaystyle \text{(iv) }G=\{x:x\text{ is an integer, }-3<x<3\}\text{ and}
\displaystyle H=\{x:x\text{ is a factor of }16\}
\displaystyle \text{Answer:}
\displaystyle \text{Yes}
\displaystyle \text{Note: }G=\{-2,-1,0,1,2\}\text{ and }H=\{1,2,4,8,16\}.
\displaystyle \text{Therefore, }n(G)=5\text{ and }n(H)=5.
\displaystyle \text{Hence }G\text{ and }H\text{ are equivalent sets.}

\displaystyle \\

\displaystyle \textbf{Question 8: }\text{State whether the following statements are true or false.}

\displaystyle \text{(i) }\{a,b,c,1,2,3\}\text{ is not a set}
\displaystyle \text{Answer:}
\displaystyle \text{False. It is a set. It contains well-defined and distinct elements.}

\displaystyle \text{(ii) }\{5,7,9\}=\{9,5,7\}
\displaystyle \text{Answer:}
\displaystyle \text{True. Both sets contain the same elements, just in a different order.}
\displaystyle \text{The order of elements in a set does not matter.}

\displaystyle \text{(iii) }\{x\mid x\in W,\ x+8=8\}\text{ is a singleton set}
\displaystyle \text{Answer:}
\displaystyle \text{True. The only element in the set is }0.\text{ Hence it is a singleton set.}
\displaystyle \text{Note: A singleton, also known as a unit set, is a set with exactly one element.}
\displaystyle \text{For example, the set }\{0\}\text{ is a singleton.}

\displaystyle \text{(iv) }\{x\mid x\in W,\ x<0\}=\phi
\displaystyle \text{Answer:}
\displaystyle \text{True. There is no whole number less than }0,\text{ so the set is an empty set.}

\displaystyle \text{(v) }\{x\mid x\in N,\ x+5=3\}=\phi
\displaystyle \text{Answer:}
\displaystyle \text{True. Since }x+5=3,\ x=-2.\text{ But }-2\notin N.
\displaystyle \text{Hence the set is an empty set.}

\displaystyle \text{(vi) }\{x\mid x\in N,\ 3<x\leq4\}=\phi
\displaystyle \text{Answer:}
\displaystyle \text{False. There is a valid element in the set, which is }4.
\displaystyle \text{Therefore, the set is }\{4\}\text{ and is not an empty set.}

\displaystyle \text{(vii) If }A=\{x\mid x\text{ is a letter of the word 'Meerut'}\},\text{ then }n(A)=6
\displaystyle \text{Answer:}
\displaystyle \text{False. }A=\{M,e,r,u,t\},\text{ since repeated elements are written only once in a set.}
\displaystyle \text{Therefore, }n(A)=5.

\displaystyle \text{(viii) If }A=\{x\mid x\in N,\ 8<x<13\}\text{ and}
\displaystyle B=\{x\mid x\in Z,\ -3\leq x<1\},\text{ then }n(A)=n(B)
\displaystyle \text{Answer:}
\displaystyle \text{True. }A=\{9,10,11,12\}.\text{ Therefore, }n(A)=4.
\displaystyle B=\{-3,-2,-1,0\}.\text{ Therefore, }n(B)=4.
\displaystyle \text{Hence }n(A)=n(B).

\displaystyle \text{(ix) If }n(A)=n(B),\text{ then }A=B
\displaystyle \text{Answer:}
\displaystyle \text{False. Just because the number of elements in sets }A\text{ and }B\text{ are equal, it does not}
\displaystyle \text{mean that the sets are equal. They are equivalent sets, but need not be equal sets.}

\displaystyle \text{(x) If }B\text{ is the set of all consonants in the English alphabet, then }n(B)=21
\displaystyle \text{Answer:}
\displaystyle \text{True.}
\displaystyle \text{Note: The }21\text{ consonant letters in the English alphabet are}
\displaystyle B,C,D,F,G,H,J,K,L,M,N,P,Q,R,S,T,V,W,X,Y,Z.

\displaystyle \text{(xi) }\{x\mid x\text{ is a prime factor of }24\}=\{2,3,4,6,8,12,24\}
\displaystyle \text{Answer:}
\displaystyle \text{False. The prime factors of }24\text{ are }2\text{ and }3.
\displaystyle \text{Therefore, }\{x\mid x\text{ is a prime factor of }24\}=\{2,3\}.

\displaystyle \\

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