\displaystyle \text{MULTIPLE CHOICE QUESTIONS (MCQs)}

\displaystyle \textbf{Question 1: }\text{For any set }A,\ (A')'\text{ is equal to}
\displaystyle \text{(a) }A'\qquad\text{(b) }A\qquad\text{(c) }\phi\qquad\text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle \text{We know that the complement of the complement of a set is the set itself.}
\displaystyle (A')'=A
\displaystyle \therefore \text{Option (b) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Let }A\text{ and }B\text{ be two sets in the same universal set. Then, } \\ A-B=
\displaystyle \text{(a) }A\cap B\qquad\text{(b) }A'\cap B\qquad\text{(c) }A\cap B'\qquad\text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle A-B=\{x:x\in A\text{ and }x\notin B\}
\displaystyle =\{x:x\in A\text{ and }x\in B'\}
\displaystyle =A\cap B'
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 3: }\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{A set containing }n\text{ elements has }2^n\text{ subsets.}
\displaystyle \therefore \text{Option (d) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{For any two sets }A\text{ and }B,\ A\cap(A\cup B)=
\displaystyle \text{(a) }A\qquad\text{(b) }B\qquad\text{(c) }\phi\qquad\text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle \text{By the absorption law,}
\displaystyle A\cap(A\cup B)=A
\displaystyle \therefore \text{Option (a) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{If }A=\{1,3,5,B\}\text{ and }B=\{2,4\},\text{ then}
\displaystyle \text{(a) }4\in A\qquad\text{(b) }\{4\}\subset A\qquad\text{(c) }B\subset A
\displaystyle \text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle A=\{1,3,5,B\}\text{ and }B=\{2,4\}.
\displaystyle \text{Since }4\notin A,\text{ option (a) is incorrect.}
\displaystyle \text{Since }4\notin A,\ \{4\}\not\subset A,\text{ so option (b) is incorrect.}
\displaystyle \text{Also, }B=\{2,4\}\text{ and neither }2\text{ nor }4\text{ is an element of }A.
\displaystyle \therefore B\not\subset A,\text{ so option (c) is incorrect.}
\displaystyle \text{Note that }B\in A,\text{ but }B\not\subset A.
\displaystyle \therefore \text{Option (d) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{The symmetric difference of }A\text{ and }B\text{ is not equal to}
\displaystyle \text{(a) }(A-B)\cap(B-A)\qquad\text{(b) }(A-B)\cup(B-A)
\displaystyle \text{(c) }(A\cup B)-(A\cap B)
\displaystyle \text{(d) }[(A\cup B)-A]\cup[(A\cup B)-B]
\displaystyle \text{Answer:}
\displaystyle \text{The symmetric difference of }A\text{ and }B\text{ is}
\displaystyle A\triangle B=(A-B)\cup(B-A)
\displaystyle \text{Also, }A\triangle B=(A\cup B)-(A\cap B)
\displaystyle [(A\cup B)-A]\cup[(A\cup B)-B]=(B-A)\cup(A-B)=A\triangle B
\displaystyle \text{But }(A-B)\cap(B-A)=\phi,\text{ since }A-B\text{ and }B-A\text{ are disjoint.}
\displaystyle \therefore \text{Option (a) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{The symmetric difference of }A=\{1,2,3\}\text{ and }B=\{3,4,5\}\text{ is}
\displaystyle \text{(a) }\{1,2\}\qquad\text{(b) }\{1,2,4,5\}\qquad\text{(c) }\{4,3\}  \qquad \text{(d) }\{2,5,1,4,3\}
\displaystyle \text{Answer:}
\displaystyle A\triangle B=(A-B)\cup(B-A)
\displaystyle A-B=\{1,2\},\qquad B-A=\{4,5\}
\displaystyle \therefore A\triangle B=\{1,2\}\cup\{4,5\}=\{1,2,4,5\}
\displaystyle \therefore \text{Option (b) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{For any two sets }A\text{ and }B,\ (A-B)\cup(B-A)=
\displaystyle \text{(a) }(A-B)\cup A\qquad\text{(b) }(B-A)\cup B
\displaystyle \text{(c) }(A\cup B)-(A\cap B)\qquad\text{(d) }(A\cup B)\cap(A\cap B)
\displaystyle \text{Answer:}
\displaystyle \text{By the definition of symmetric difference,}
\displaystyle (A-B)\cup(B-A)=A\triangle B
\displaystyle \text{Also, }A\triangle B=(A\cup B)-(A\cap B)
\displaystyle \therefore (A-B)\cup(B-A)=(A\cup B)-(A\cap B)
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Which of the following statements is false?}
\displaystyle \text{(a) }A-B=A\cap B'\qquad\text{(b) }A-B=A-(A\cap B)
\displaystyle \text{(c) }A-B=A-B'\qquad\text{(d) }A-B=(A\cup B)-B
\displaystyle \text{Answer:}
\displaystyle \text{We know that }A-B=A\cap B'.
\displaystyle A-(A\cap B)=A\cap(A\cap B)'
\displaystyle =A\cap(A'\cup B')=(A\cap A')\cup(A\cap B')=A\cap B'=A-B
\displaystyle \text{Also, }(A\cup B)-B=(A\cup B)\cap B'
\displaystyle =(A\cap B')\cup(B\cap B')=A\cap B'=A-B
\displaystyle \text{But }A-B'=A\cap(B')'=A\cap B,\text{ which is not equal to }A-B\text{ in general.}
\displaystyle \therefore \text{Option (c) is the false statement.}
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{For any three sets }A,\ B\text{ and }C,
\displaystyle \text{(a) }A\cap(B-C)=(A\cap B)-(A\cap C)
\displaystyle \text{(b) }A\cap(B-C)=(A\cap B)-C
\displaystyle \text{(c) }A\cup(B-C)=(A\cup B)\cap(A\cup C')
\displaystyle \text{(d) }A\cup(B-C)=(A\cup B)-(A\cup C)
\displaystyle \text{Answer:}
\displaystyle A\cap(B-C)=A\cap(B\cap C')
\displaystyle =(A\cap B)\cap C'
\displaystyle \text{Also, }(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 =A\cap B\cap C'
\displaystyle \therefore A\cap(B-C)=(A\cap B)-(A\cap C)
\displaystyle \therefore \text{Option (a) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Let }A=\{x:x\in R,\ x>4\}\text{ and } \\ B=\{x\in R:x<5\}.\text{ Then, }A\cap B=
\displaystyle \text{(a) }(4,5]\qquad\text{(b) }(4,5)\qquad\text{(c) }[4,5)\qquad\text{(d) }[4,5]
\displaystyle \text{Answer:}
\displaystyle A=\{x\in R:x>4\}=(4,\infty)
\displaystyle B=\{x\in R:x<5\}=(-\infty,5)
\displaystyle A\cap B=\{x\in R:4<x<5\}
\displaystyle \therefore A\cap B=(4,5)
\displaystyle \therefore \text{Option (b) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{Let }U\text{ be the universal set containing }700\text{ elements. If }A,B\text{ are}
\displaystyle \text{subsets of }U\text{ such that }n(A)=200,\ n(B)=300\text{ and }n(A\cap B)=100.
\displaystyle \text{Then, }n(A'\cap B')=
\displaystyle \text{(a) }400\qquad\text{(b) }600\qquad\text{(c) }300\qquad\text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)
\displaystyle =200+300-100=400
\displaystyle \text{By De Morgan's law, }A'\cap B'=(A\cup B)'
\displaystyle \therefore n(A'\cap B')=n(U)-n(A\cup B)
\displaystyle =700-400=300
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{Let }A\text{ and }B\text{ be two sets such that }n(A)=16,\ n(B)=14,
\displaystyle n(A\cup B)=25.\text{ Then, }n(A\cap B)\text{ is equal to}
\displaystyle \text{(a) }30\qquad\text{(b) }50\qquad\text{(c) }5\qquad\text{(d) none of these}
\displaystyle \text{Answer:}
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)
\displaystyle 25=16+14-n(A\cap B)
\displaystyle n(A\cap B)=30-25=5
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{If }A=\{1,2,3,4,5\},\text{ then the number of proper subsets of }A\text{ is}
\displaystyle \text{(a) }120\qquad\text{(b) }30\qquad\text{(c) }31\qquad\text{(d) }32
\displaystyle \text{Answer:}
\displaystyle \text{Since }A\text{ has }5\text{ elements, the total number of subsets is }2^5=32.
\displaystyle \text{A proper subset of }A\text{ is any subset of }A\text{ other than }A\text{ itself.}
\displaystyle \therefore \text{Number of proper subsets}=2^5-1=32-1=31
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{In set-builder method the null set is represented by}
\displaystyle \text{(a) }\{\}\qquad\text{(b) }\phi\qquad\text{(c) }\{x:x\ne x\}\qquad\text{(d) }\{x:x=x\}
\displaystyle \text{Answer:}
\displaystyle \text{In set-builder form, the null set can be represented as }\{x:x\ne x\}.
\displaystyle \text{Since no element can be unequal to itself, this set contains no elements.}
\displaystyle \therefore \{x:x\ne x\}=\phi
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{If }A\text{ and }B\text{ are two disjoint sets, then }n(A\cup B)\text{ is equal to}
\displaystyle \text{(a) }n(A)+n(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)
\displaystyle \text{Answer:}
\displaystyle \text{Since }A\text{ and }B\text{ are disjoint sets, }A\cap B=\phi.
\displaystyle \therefore n(A\cap B)=0
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)
\displaystyle =n(A)+n(B)
\displaystyle \therefore \text{Option (a) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{For two sets }A\cup B=A\text{ iff}
\displaystyle \text{(a) }B\subseteq A\qquad\text{(b) }A\subseteq B\qquad\text{(c) }A\ne B\qquad\text{(d) }A=B
\displaystyle \text{Answer:}
\displaystyle A\cup B=A\text{ iff every element of }B\text{ is already an element of }A.
\displaystyle \therefore A\cup B=A\iff B\subseteq A
\displaystyle \therefore \text{Option (a) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{If }A\text{ and }B\text{ are two sets such that }n(A)=70,\ n(B)=60,
\displaystyle n(A\cup B)=110,\text{ then }n(A\cap B)\text{ is equal to}
\displaystyle \text{(a) }240\qquad\text{(b) }50\qquad\text{(c) }40\qquad\text{(d) }20
\displaystyle \text{Answer:}
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)
\displaystyle 110=70+60-n(A\cap B)
\displaystyle n(A\cap B)=130-110=20
\displaystyle \therefore \text{Option (d) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{If }A\text{ and }B\text{ are two given sets, then }A\cap(A\cap B)'\text{ is equal to}
\displaystyle \text{(a) }A\qquad\text{(b) }B\qquad\text{(c) }\phi\qquad\text{(d) }A\cap B^c
\displaystyle \text{Answer:}
\displaystyle A\cap(A\cap B)'=A\cap(A'\cup B')
\displaystyle =(A\cap A')\cup(A\cap B')
\displaystyle =\phi\cup(A\cap B')=A\cap B'
\displaystyle \therefore A\cap(A\cap B)'=A\cap B^c
\displaystyle \therefore \text{Option (d) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{If }A=\{x:x\text{ is a multiple of }3\}\text{ and}
\displaystyle B=\{x:x\text{ is a multiple of }5\},\text{ then }A-B\text{ is}
\displaystyle \text{(a) }A\cap B\qquad\text{(b) }A\cap\overline{B}\qquad\text{(c) }\overline{A}\cap\overline{B}\qquad\text{(d) }\overline{A}\cap B
\displaystyle \text{Answer:}
\displaystyle A-B=\{x:x\in A\text{ and }x\notin B\}
\displaystyle =A\cap B'=A\cap\overline{B}
\displaystyle \therefore \text{Option (b) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 21: }\text{In a city, }20\%\text{ of the population travels by car, }50\%\text{ travels}
\displaystyle \text{by bus and }10\%\text{ travels by both car and bus. Then, persons travelling by car}
\displaystyle \text{or bus is}
\displaystyle \text{(a) }80\%\qquad\text{(b) }40\%\qquad\text{(c) }60\%\qquad\text{(d) }70\%
\displaystyle \text{Answer:}
\displaystyle \text{Let }A\text{ be the set of persons travelling by car and }B\text{ by bus.}
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)
\displaystyle =20\%+50\%-10\%=60\%
\displaystyle \therefore \text{The percentage of persons travelling by car or bus is }60\%.
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 22: }\text{If }A\cap B=B,\text{ then}
\displaystyle \text{(a) }A\subseteq B\qquad\text{(b) }B\subseteq A\qquad\text{(c) }A=\phi\qquad\text{(d) }B=\phi
\displaystyle \text{Answer:}
\displaystyle A\cap B=B
\displaystyle \text{This means that every element of }B\text{ is also an element of }A.
\displaystyle \therefore B\subseteq A
\displaystyle \therefore \text{Option (b) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 23: }\text{An investigator interviewed }100\text{ students to determine the performance}
\displaystyle \text{of three drinks: milk, coffee and tea. The investigator reported that }10\text{ students}
\displaystyle \text{take all three drinks milk, coffee and tea; }20\text{ students take milk and coffee;}
\displaystyle 25\text{ students take milk and tea; }20\text{ students take coffee and tea; }12\text{ students}
\displaystyle \text{take milk only; }5\text{ students take coffee only and }8\text{ students take tea only.}
\displaystyle \text{Then the number of students who did not take any of three drinks is}
\displaystyle \text{(a) }10\qquad\text{(b) }20\qquad\text{(c) }25\qquad\text{(d) }30
\displaystyle \text{Answer:}
\displaystyle \text{Let }M,\ C\text{ and }T\text{ denote the sets of students taking milk, coffee and tea.}
\displaystyle n(M\cap C\cap T)=10
\displaystyle \text{Students taking milk and coffee only}=20-10=10
\displaystyle \text{Students taking milk and tea only}=25-10=15
\displaystyle \text{Students taking coffee and tea only}=20-10=10
\displaystyle \text{Students taking at least one drink}
\displaystyle =12+5+8+10+15+10+10
\displaystyle =70
\displaystyle \therefore \text{Students who did not take any of the three drinks}=100-70=30
\displaystyle \therefore \text{Option (d) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 24: }\text{Two finite sets have }m\text{ and }n\text{ elements. The number of elements in the}
\displaystyle \text{power set of first set is }48\text{ more than the total number of elements in power}
\displaystyle \text{set of the second set. Then, the values 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{The number of elements in the power sets are }2^m\text{ and }2^n.
\displaystyle \therefore 2^m-2^n=48
\displaystyle \text{Checking option (c), }m=6,\ n=4,
\displaystyle 2^6-2^4=64-16=48
\displaystyle \therefore m=6,\ n=4
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 25: }\text{In a class of }175\text{ students the following data shows the number of}
\displaystyle \text{students opting one or more subjects. Mathematics }100;\text{ Physics }70;\text{ Chemistry }40;
\displaystyle \text{Mathematics and Physics }30;\text{ Mathematics and Chemistry }28;\text{ Physics and}
\displaystyle \text{Chemistry }23;\text{ Mathematics, Physics and Chemistry }18.\text{ How many students}
\displaystyle \text{have offered Mathematics alone?}
\displaystyle \text{(a) }35\qquad\text{(b) }48\qquad\text{(c) }60\qquad\text{(d) }22
\displaystyle \text{Answer:}
\displaystyle \text{Students taking Mathematics and Physics only}=30-18=12
\displaystyle \text{Students taking Mathematics and Chemistry only}=28-18=10
\displaystyle \text{Students taking all three subjects}=18
\displaystyle \text{Let the number of students taking Mathematics alone}=x.
\displaystyle x+12+10+18=100
\displaystyle x=100-40=60
\displaystyle \therefore \text{The number of students who offered Mathematics alone is }60.
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 26: }\text{Suppose }A_1,A_2,\ldots,A_{30}\text{ are thirty sets each having }5\text{ elements}
\displaystyle \text{and }B_1,B_2,\ldots,B_n\text{ are }n\text{ sets each with }3\text{ elements, let}
\displaystyle \bigcup_{i=1}^{30}A_i=\bigcup_{j=1}^{n}B_j=S\text{ and each element of }S\text{ belongs to exactly }10\text{ of the }A_i\text{'s}
\displaystyle \text{and exactly }9\text{ of the }B_j\text{'s, then }n\text{ is equal to}
\displaystyle \text{(a) }15\qquad\text{(b) }3\qquad\text{(c) }45\qquad\text{(d) }35
\displaystyle \text{Answer:}
\displaystyle \text{Total number of memberships in }A_1,A_2,\ldots,A_{30}=30\times5=150.
\displaystyle \text{Each element of }S\text{ belongs to exactly }10\text{ of the }A_i\text{'s.}
\displaystyle \therefore 10n(S)=150
\displaystyle \therefore n(S)=15
\displaystyle \text{Total number of memberships in }B_1,B_2,\ldots,B_n=3n.
\displaystyle \text{Each element of }S\text{ belongs to exactly }9\text{ of the }B_j\text{'s.}
\displaystyle \therefore 3n=9n(S)=9\times15=135
\displaystyle \therefore n=45
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 27: }\text{Two finite sets have }m\text{ and }n\text{ elements. The number of subsets of}
\displaystyle \text{the first set is }112\text{ more than that of the second. The values of }m\text{ and }n
\displaystyle \text{are respectively}
\displaystyle \text{(a) }4,7\qquad\text{(b) }7,4\qquad\text{(c) }4,4\qquad\text{(d) }7,7
\displaystyle \text{Answer:}
\displaystyle \text{The numbers of subsets of the two sets are }2^m\text{ and }2^n\text{ respectively.}
\displaystyle \therefore 2^m-2^n=112
\displaystyle \text{For }m=7\text{ and }n=4,
\displaystyle 2^7-2^4=128-16=112
\displaystyle \therefore m=7,\ n=4
\displaystyle \therefore \text{Option (b) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 28: }\text{For any two sets }A\text{ and }B,\ A\cap(A\cup B)'\text{ is equal to}
\displaystyle \text{(a) }A\qquad\text{(b) }B\qquad\text{(c) }\phi\qquad\text{(d) }A\cap B
\displaystyle \text{Answer:}
\displaystyle \text{By De Morgan's law, }(A\cup B)'=A'\cap B'.
\displaystyle \therefore A\cap(A\cup B)'=A\cap A'\cap B'
\displaystyle =\phi\cap B'=\phi
\displaystyle \therefore \text{Option (c) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 29: }\text{The set }(A\cup B')\cup(B\cap C)\text{ is equal to}
\displaystyle \text{(a) }A'\cup B\cup C\qquad\text{(b) }A\cup B'\cup C
\displaystyle \text{(c) }A'\cup C'\qquad\text{(d) }A'\cap B
\displaystyle \text{Answer:}
\displaystyle (A\cup B')\cup(B\cap C)=A\cup[B'\cup(B\cap C)]
\displaystyle =A\cup[(B'\cup B)\cap(B'\cup C)]
\displaystyle =A\cup[U\cap(B'\cup C)]
\displaystyle =A\cup B'\cup C
\displaystyle \therefore \text{Option (b) is correct.}
\displaystyle \\

\displaystyle \textbf{Question 30: }\text{Let }F_1\text{ be the set of all parallelograms, }F_2\text{ the set of all rectangles,}
\displaystyle F_3\text{ the set of all rhombuses, }F_4\text{ the set of all squares and }F_5\text{ the set of}
\displaystyle \text{trapeziums in a plane. Then }F_1\text{ may be equal to}
\displaystyle \text{(a) }F_2\cap F_3\qquad\text{(b) }F_3\cap F_4\qquad\text{(c) }F_2\cup F_3
\displaystyle \text{(d) }F_2\cup F_3\cup F_4\cup F_1
\displaystyle \text{Answer:}
\displaystyle F_2\subseteq F_1,\qquad F_3\subseteq F_1,\qquad F_4\subseteq F_1
\displaystyle \therefore F_2\cup F_3\cup F_4\cup F_1=F_1
\displaystyle \therefore \text{Option (d) is correct.}
\displaystyle \\

\displaystyle \text{VERY SHORT ANSWER QUESTIONS}

\displaystyle \textbf{Question 1: }\text{If a set contains }n\text{ elements, then write the number of elements in its}
\displaystyle \text{power set.}
\displaystyle \text{Answer:}
\displaystyle \text{A set containing }n\text{ elements has }2^n\text{ subsets.}
\displaystyle \therefore \text{The number of elements in its power set is }2^n.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Write the number of elements in the power set of null set.}
\displaystyle \text{Answer:}
\displaystyle n(\phi)=0
\displaystyle \therefore n(P(\phi))=2^0=1
\displaystyle \text{In fact, }P(\phi)=\{\phi\}.
\displaystyle \therefore \text{The number of elements in the power set of null set is }1.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Let }A=\{x:x\in N,\ x\text{ is a multiple of }3\}\text{ and}
\displaystyle B=\{x:x\in N\text{ and }x\text{ is a multiple of }5\}.\text{ Write }A\cap B.
\displaystyle \text{Answer:}
\displaystyle \text{The elements of }A\cap B\text{ are multiples of both }3\text{ and }5.
\displaystyle \mathrm{LCM}(3,5)=15
\displaystyle \therefore A\cap B=\{15,30,45,60,\ldots\}
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Let }A\text{ and }B\text{ be two sets having }3\text{ and }6\text{ elements respectively.}
\displaystyle \text{Write the minimum number of elements that }A\cup B\text{ can have.}
\displaystyle \text{Answer:}
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)
\displaystyle =3+6-n(A\cap B)
\displaystyle \text{For }n(A\cup B)\text{ to be minimum, }n(A\cap B)\text{ must be maximum.}
\displaystyle \text{The maximum possible value of }n(A\cap B)\text{ is }3.
\displaystyle \therefore \text{Minimum }n(A\cup B)=3+6-3=6
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{If }A=\{x\in C:x^2=1\}\text{ and }B=\{x\in C:x^4=1\},\text{ then write}
\displaystyle A-B\text{ and }B-A.
\displaystyle \text{Answer:}
\displaystyle x^2=1\Rightarrow x=\pm1
\displaystyle \therefore A=\{-1,1\}
\displaystyle x^4=1\Rightarrow x=\pm1,\ \pm i
\displaystyle \therefore B=\{-1,1,i,-i\}
\displaystyle \therefore A-B=\phi\text{ and }B-A=\{i,-i\}
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{If }A\text{ and }B\text{ are two sets such that }A\subset B,\text{ then write }B'-A'
\displaystyle \text{in terms of }A\text{ and }B.
\displaystyle \text{Answer:}
\displaystyle B'-A'=B'\cap(A')'
\displaystyle =B'\cap A
\displaystyle =A-B
\displaystyle \text{Since }A\subset B,\ A-B=\phi.
\displaystyle \therefore B'-A'=A-B=\phi
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Let }A\text{ and }B\text{ be two sets having }4\text{ and }7\text{ elements respectively.}
\displaystyle \text{Then write the maximum number of elements that }A\cup B\text{ can have.}
\displaystyle \text{Answer:}
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)
\displaystyle \text{For }n(A\cup B)\text{ to be maximum, }n(A\cap B)\text{ must be minimum.}
\displaystyle \text{The minimum possible value of }n(A\cap B)\text{ is }0.
\displaystyle \therefore \text{Maximum }n(A\cup B)=4+7=11
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{If }A=\{(x,y):y=\frac{1}{x},\ 0\ne x\in R\}\text{ and}
\displaystyle B=\{(x,y):y=-x,\ x\in R\},\text{ then write }A\cap B.
\displaystyle \text{Answer:}
\displaystyle \text{For }(x,y)\in A\cap B,\quad \frac{1}{x}=-x.
\displaystyle \therefore x^2=-1
\displaystyle \text{Since }x\in R,\text{ the equation }x^2=-1\text{ has no solution.}
\displaystyle \therefore A\cap B=\phi
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{If }A=\{(x,y):y=e^x,\ x\in R\}\text{ and}
\displaystyle B=\{(x,y):y=e^{-x},\ x\in R\},\text{ then write }A\cap B.
\displaystyle \text{Answer:}
\displaystyle \text{For }(x,y)\in A\cap B,\quad e^x=e^{-x}.
\displaystyle \therefore e^{2x}=1
\displaystyle \therefore x=0
\displaystyle \text{When }x=0,\quad y=e^0=1.
\displaystyle \therefore A\cap B=\{(0,1)\}
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{If }A\text{ and }B\text{ are two sets such that }n(A)=20,\ n(B)=25
\displaystyle \text{and }n(A\cup B)=40,\text{ then write }n(A\cap B).
\displaystyle \text{Answer:}
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)
\displaystyle 40=20+25-n(A\cap B)
\displaystyle \therefore n(A\cap B)=5
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{If }A\text{ and }B\text{ are two sets such that }n(A)=115,\ n(B)=326,
\displaystyle n(A-B)=47,\text{ then write }n(A\cup B).
\displaystyle \text{Answer:}
\displaystyle n(A-B)=n(A)-n(A\cap B)
\displaystyle 47=115-n(A\cap B)
\displaystyle \therefore n(A\cap B)=68
\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)
\displaystyle =115+326-68=373
\displaystyle \therefore n(A\cup B)=373
\displaystyle \\

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