\displaystyle \textbf{Question 1: If }A\text{ and }B\text{ are two sets such that }A\subset B,\text{ find:}
\displaystyle \text{(i) }A\cap B\qquad\text{(ii) }A\cup B
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) Since }A\subset B,\text{ every element of }A\text{ belongs to }B.
\displaystyle \therefore A\cap B=A.
\displaystyle \text{(ii) Since }B\text{ already contains all the elements of }A,
\displaystyle \therefore A\cup B=B.
\displaystyle \\

\displaystyle \textbf{Question 2: If }A=\{1,2,3,4,5\},\;B=\{4,5,6,7,8\},\;C=\{7,8,9,10,11\}
\displaystyle \text{and }D=\{10,11,12,13,14\},\text{ find:}
\displaystyle \text{(i) }A\cup B\quad\text{(ii) }A\cup C\quad\text{(iii) }B\cup C\quad\text{(iv) }B\cup D
\displaystyle \text{(v) }A\cup B\cup C\quad\text{(vi) }A\cup B\cup D\quad\text{(vii) }B\cup C\cup D
\displaystyle \text{(viii) }A\cap(B\cup C)\quad\text{(ix) }(A\cap B)\cap(B\cap C)\quad\text{(x) }(A\cup D)\cap(B\cup C)
\displaystyle \textbf{Answer:}
\displaystyle \text{Given }A=\{1,2,3,4,5\},\;B=\{4,5,6,7,8\},\;C=\{7,8,9,10,11\},\;D=\{10,11,12,13,14\}.
\displaystyle \text{(i) }A\cup B=\{1,2,3,4,5,6,7,8\}.
\displaystyle \text{(ii) }A\cup C=\{1,2,3,4,5,7,8,9,10,11\}.
\displaystyle \text{(iii) }B\cup C=\{4,5,6,7,8,9,10,11\}.
\displaystyle \text{(iv) }B\cup D=\{4,5,6,7,8,10,11,12,13,14\}.
\displaystyle \text{(v) }A\cup B\cup C=\{1,2,3,4,5,6,7,8,9,10,11\}.
\displaystyle \text{(vi) }A\cup B\cup D=\{1,2,3,4,5,6,7,8,10,11,12,13,14\}.
\displaystyle \text{(vii) }B\cup C\cup D=\{4,5,6,7,8,9,10,11,12,13,14\}.
\displaystyle \text{(viii) }A\cap(B\cup C)=\{4,5\}.
\displaystyle \text{(ix) }(A\cap B)\cap(B\cap C)=\phi.
\displaystyle \text{(x) }(A\cup D)\cap(B\cup C)=\{4,5,10,11\}.
\displaystyle \\

\displaystyle \textbf{Question 3: Let }A=\{x:x\in N\},\;B=\{x:x=2n,\;n\in N\},
\displaystyle C=\{x:x=2n-1,\;n\in N\},\;D=\{x:x\text{ is a prime natural number}\}.
\displaystyle \text{Find: (i) }A\cap B\;\text{(ii) }A\cap C\;\text{(iii) }A\cap D\;\text{(iv) }B\cap C
\displaystyle \text{(v) }B\cap D\;\text{(vi) }C\cap D
\displaystyle \textbf{Answer:}
\displaystyle A=\{1,2,3,4,5,\ldots\}.
\displaystyle B=\{2,4,6,8,10,\ldots\}.
\displaystyle C=\{1,3,5,7,9,\ldots\}.
\displaystyle D=\{2,3,5,7,11,13,17,19,23,\ldots\}.
\displaystyle \text{(i) }A\cap B=\{2,4,6,8,10,\ldots\}=B.
\displaystyle \text{(ii) }A\cap C=\{1,3,5,7,9,\ldots\}=C.
\displaystyle \text{(iii) }A\cap D=\{2,3,5,7,11,13,17,19,23,\ldots\}=D.
\displaystyle \text{(iv) }B\cap C=\phi.
\displaystyle \text{(v) }B\cap D=\{2\}.
\displaystyle \text{(vi) }C\cap D=\{3,5,7,11,13,17,19,23,\ldots\}=D-\{2\}.
\displaystyle \\

\displaystyle \textbf{Question 4: Let }A=\{3,6,12,15,18,21\},\;B=\{4,8,12,16,20\},
\displaystyle C=\{2,4,6,8,10,12,14,16\}\text{ and }D=\{5,10,15,20\}. \text{ Find:}
\displaystyle \text{(i) }A-B\quad\text{(ii) }A-C\quad\text{(iii) }A-D\quad\text{(iv) }B-A
\displaystyle \text{(v) }C-A\quad\text{(vi) }D-A\quad\text{(vii) }B-C\quad\text{(viii) }B-D
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) }A-B=\{3,6,15,18,21\}.
\displaystyle \text{(ii) }A-C=\{3,15,18,21\}.
\displaystyle \text{(iii) }A-D=\{3,6,12,18,21\}.
\displaystyle \text{(iv) }B-A=\{4,8,16,20\}.
\displaystyle \text{(v) }C-A=\{2,4,8,10,14,16\}.
\displaystyle \text{(vi) }D-A=\{5,10,20\}.
\displaystyle \text{(vii) }B-C=\{20\}.
\displaystyle \text{(viii) }B-D=\{4,8,12,16\}.
\displaystyle \\

\displaystyle \textbf{Question 5: If }U=\{1,2,3,4,5,6,7,8,9\},\;A=\{1,2,3,4\},
\displaystyle B=\{2,4,6,8\}\text{ and }C=\{3,4,5,6\},\text{ find:}
\displaystyle \text{(i) }A'\quad\text{(ii) }B'\quad\text{(iii) }(A\cap C)'\quad\text{(iv) }(A\cup B)'
\displaystyle \text{(v) }(A')'\quad\text{(vi) }(B-C)'
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) }A'=\{5,6,7,8,9\}.
\displaystyle \text{(ii) }B'=\{1,3,5,7,9\}.
\displaystyle \text{(iii) }A\cap C=\{3,4\}.
\displaystyle \therefore (A\cap C)'=\{1,2,5,6,7,8,9\}.
\displaystyle \text{(iv) }A\cup B=\{1,2,3,4,6,8\}.
\displaystyle \therefore (A\cup B)'=\{5,7,9\}.
\displaystyle \text{(v) }(A')'=\{1,2,3,4\}=A.
\displaystyle \text{(vi) }B-C=\{2,8\}.
\displaystyle \therefore (B-C)'=\{1,3,4,5,6,7,9\}.
\displaystyle \\

\displaystyle \textbf{Question 6: Let }U=\{1,2,3,4,5,6,7,8,9\},\;A=\{2,4,6,8\}\text{ and }B=\{2,3,5,7\}.
\displaystyle \text{Verify that: (i) }(A\cup B)'=A'\cap B'\qquad\text{(ii) }(A\cap B)'=A'\cup B'.
\displaystyle \textbf{Answer:}
\displaystyle \text{(i) }A\cup B=\{2,3,4,5,6,7,8\}.
\displaystyle \therefore (A\cup B)'=\{1,9\}.
\displaystyle A'=\{1,3,5,7,9\},\qquad B'=\{1,4,6,8,9\}.
\displaystyle A'\cap B'=\{1,9\}.
\displaystyle \therefore (A\cup B)'=A'\cap B'.
\displaystyle \text{(ii) }A\cap B=\{2\}.
\displaystyle \therefore (A\cap B)'=\{1,3,4,5,6,7,8,9\}.
\displaystyle A'=\{1,3,5,7,9\},\qquad B'=\{1,4,6,8,9\}.
\displaystyle A'\cup B'=\{1,3,4,5,6,7,8,9\}.
\displaystyle \therefore (A\cap B)'=A'\cup B'.
\displaystyle \\

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