\displaystyle \textbf{Question 1: Which of the following are examples of empty set?}
\displaystyle \text{(i) Set of all even natural numbers divisible by }5.
\displaystyle \text{(ii) Set of all even prime numbers.}
\displaystyle \text{(iii) }\{x:x^2-2=0\text{ and }x\text{ is rational}\}.
\displaystyle \text{(iv) }\{x:x\text{ is a natural number, }x<8\text{ and }x>12\}.
\displaystyle \text{(v) }\{x:x\text{ is a point common to any two parallel lines}\}.
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) The set of all even natural numbers divisible by }5\text{ is }
\displaystyle \{x:x=10n,\;n\in N\}=\{10,20,30,40,\ldots\}.
\displaystyle \text{This set has elements. Hence, it is not an empty set.}
\displaystyle \text{(ii) The set of all even prime numbers is }\{2\}.
\displaystyle \text{This set has an element. Hence, it is not an empty set.}
\displaystyle \text{(iii) From }x^2-2=0,\;x=\pm\sqrt{2}.
\displaystyle \sqrt{2}\text{ is irrational. Hence, there is no rational value of }x.
\displaystyle \therefore \text{This is an empty set.}
\displaystyle \text{(iv) There is no natural number which is less than }8
\displaystyle \text{and greater than }12\text{ simultaneously.}
\displaystyle \therefore \text{This is an empty set.}
\displaystyle \text{(v) Two parallel lines never intersect.}
\displaystyle \therefore \text{There is no common point. Hence, this is an empty set.}
\displaystyle \therefore \text{The empty sets are (iii), (iv) and (v).}
\displaystyle \\

\displaystyle \textbf{Question 2: Which of the following sets are finite and which are infinite?}
\displaystyle \text{(i) Set of concentric circles in a plane.}
\displaystyle \text{(ii) Set of letters of the English alphabet.}
\displaystyle \text{(iii) }\{x\in N:x>5\}.
\displaystyle \text{(iv) }\{x\in N:x<200\}.
\displaystyle \text{(v) }\{x\in Z:x<5\}.
\displaystyle \text{(vi) }\{x\in R:0<x<1\}.
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) There can be infinitely many concentric circles in a plane.}
\displaystyle \therefore \text{This is an INFINITE set.}
\displaystyle \text{(ii) The English alphabet has }26\text{ letters:}
\displaystyle \{A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z\}.
\displaystyle \therefore \text{This is a FINITE set.}
\displaystyle \text{(iii) }\{x\in N:x>5\}=\{6,7,8,\ldots\}.
\displaystyle \text{This set has infinitely many elements.}
\displaystyle \therefore \text{This is an INFINITE set.}
\displaystyle \text{(iv) }\{x\in N:x<200\}=\{1,2,3,\ldots,198,199\}.
\displaystyle \text{This set has a finite number of elements.}
\displaystyle \therefore \text{This is a FINITE set.}
\displaystyle \text{(v) }\{x\in Z:x<5\}=\{\ldots,-4,-3,-2,-1,0,1,2,3,4\}.
\displaystyle \text{This set has infinitely many elements.}
\displaystyle \therefore \text{This is an INFINITE set.}
\displaystyle \text{(vi) }\{x\in R:0<x<1\}\text{ contains infinitely many real numbers.}
\displaystyle \therefore \text{This is an INFINITE set.}
\displaystyle \\

\displaystyle \textbf{Question 3: Which of the following sets are equal?}
\displaystyle \text{(i) }A=\{1,2,3\}
\displaystyle \text{(ii) }B=\{x\in R:x^2-2x+1=0\}
\displaystyle \text{(iii) }C=\{1,2,2,3\}
\displaystyle \text{(iv) }D=\{x\in R:x^3-6x^2+11x-6=0\}
\displaystyle \textbf{Answer:}
\displaystyle \text{Two sets are equal if they contain exactly the same elements.}
\displaystyle \text{For }B,\;x^2-2x+1=0
\displaystyle (x-1)^2=0\Rightarrow x=1.
\displaystyle \therefore B=\{1\}.
\displaystyle \text{For }C,\;\{1,2,2,3\}=\{1,2,3\}\text{ since repeated elements are ignored.}
\displaystyle \text{For }D,\;x^3-6x^2+11x-6=0
\displaystyle (x-1)(x-2)(x-3)=0.
\displaystyle \therefore D=\{1,2,3\}.
\displaystyle \therefore A=C=D\text{ and }B\ne A.
\displaystyle \\

\displaystyle \textbf{Question 4: Are the following sets equal?}
\displaystyle \text{(i) }A=\{x:x\text{ is a letter in the word reap}\}
\displaystyle \text{(ii) }B=\{x:x\text{ is a letter in the word paper}\}
\displaystyle \text{(iii) }C=\{x:x\text{ is a letter in the word rope}\}
\displaystyle \textbf{Answer:}
\displaystyle A=\{r,e,a,p\}.
\displaystyle B=\{p,a,e,r\}=\{r,e,a,p\}.
\displaystyle C=\{r,o,p,e\}.
\displaystyle \therefore A=B\ne C.
\displaystyle \\

\displaystyle \textbf{Question 5: From the sets given below, find the pair of equivalent sets.}
\displaystyle A=\{1,2,3\},\;B=\{t,p,q,r,s\},\;C=\{\alpha,\beta,\gamma\},\;D=\{a,e,i,o,u\}
\displaystyle \textbf{Answer:}
\displaystyle \text{Two finite sets are equivalent if they have the same cardinal number, i.e., }n(A)=n(B).
\displaystyle n(A)=3,\qquad n(B)=5,
\displaystyle n(C)=3,\qquad n(D)=5.
\displaystyle \therefore A\text{ and }C\text{ are equivalent sets.}
\displaystyle \therefore B\text{ and }D\text{ are equivalent sets.}
\displaystyle \\

\displaystyle \textbf{Question 6: Are the following pairs of sets equal? Give reasons.}
\displaystyle \text{(i) }A=\{2,3\},\;B=\{x:x\text{ is a solution of }x^2+5x+6=0\}
\displaystyle \text{(ii) }A=\{x:x\text{ is a letter of the word 'WOLF'}\},
\displaystyle \qquad\;\;B=\{x:x\text{ is a letter of the word 'FOLLOW'}\}
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) }A=\{2,3\}.
\displaystyle x^2+5x+6=0\Rightarrow(x+2)(x+3)=0.
\displaystyle \therefore B=\{-2,-3\}.
\displaystyle \therefore A\ne B.
\displaystyle \text{(ii) }A=\{W,O,L,F\}.
\displaystyle B=\{F,O,L,W\}=\{W,O,L,F\}.
\displaystyle \therefore A=B.
\displaystyle \\

\displaystyle \textbf{Question 7: From the sets given below, select the equal sets and equivalent sets.}
\displaystyle A=\{0,a\},\;B=\{1,2,3,4\},\;C=\{4,8,12\},\;D=\{3,1,2,4\},
\displaystyle E=\{1,0\},\;F=\{8,4,12\},\;G=\{1,5,7,11\},\;H=\{a,b\}
\displaystyle \textbf{Answer:}
\displaystyle n(A)=2,\qquad n(B)=4,\qquad n(C)=3,\qquad n(D)=4.
\displaystyle n(E)=2,\qquad n(F)=3,\qquad n(G)=4,\qquad n(H)=2.
\displaystyle B=\{1,2,3,4\}\text{ and }D=\{3,1,2,4\}.
\displaystyle \therefore B=D.
\displaystyle C=\{4,8,12\}\text{ and }F=\{8,4,12\}.
\displaystyle \therefore C=F.
\displaystyle \therefore \text{The equal sets are }B,D\text{ and }C,F.
\displaystyle n(A)=n(E)=n(H)=2.
\displaystyle \therefore A,E\text{ and }H\text{ are equivalent sets.}
\displaystyle n(C)=n(F)=3.
\displaystyle \therefore C\text{ and }F\text{ are equivalent sets.}
\displaystyle n(B)=n(D)=n(G)=4.
\displaystyle \therefore B,D\text{ and }G\text{ are equivalent sets.}
\displaystyle \\

\displaystyle \textbf{Question 8: Which of the following sets are equal?}
\displaystyle A=\{x:x\in N,\;x<3\},\quad B=\{1,2\},\quad C=\{3,1\},
\displaystyle D=\{x:x\in N,\;x\text{ is odd and }x<5\},
\displaystyle E=\{1,2,1,1\},\quad F=\{1,1,3\}
\displaystyle \textbf{Answer:}
\displaystyle A=\{x:x\in N,\;x<3\}=\{1,2\}.
\displaystyle B=\{1,2\}.
\displaystyle C=\{3,1\}=\{1,3\}.
\displaystyle D=\{x:x\in N,\;x\text{ is odd and }x<5\}=\{1,3\}.
\displaystyle E=\{1,2,1,1\}=\{1,2\}.
\displaystyle F=\{1,1,3\}=\{1,3\}.
\displaystyle \therefore A=B=E\text{ and }C=D=F.
\displaystyle n(A)=n(B)=n(C)=n(D)=n(E)=n(F)=2.
\displaystyle \therefore \text{All the given sets are equivalent sets.}
\displaystyle \\

\displaystyle \textbf{Question 9: Show that the set of letters needed to spell "CATARACT"}
\displaystyle \text{and the set of letters needed to spell "TRACT" are equal.}
\displaystyle \textbf{Answer:}
\displaystyle \text{The set of letters needed to spell "CATARACT" is }\{C,A,T,R\}.
\displaystyle \text{(Repeated letters are written only once in a set.)}
\displaystyle \text{The set of letters needed to spell "TRACT" is }\{T,R,A,C\}.
\displaystyle \text{Both sets contain exactly the same elements.}
\displaystyle \therefore \{C,A,T,R\}=\{T,R,A,C\}.
\displaystyle \therefore \text{The given sets are equal.}
\displaystyle \\

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