Class 11: Sets – Very Short Answer Questions

$\displaystyle \textbf{Question 1. }\text{If a set contains }n\text{ elements, then write the number of elements} \\ \text{in its power set.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Number of elements in the power set}=2^n$
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$\displaystyle \textbf{Question 2. }\text{Write the number of elements in the power set of null set.}$
$\displaystyle \text{Answer:}$
$\displaystyle P(\phi)=\{\phi\}$
$\displaystyle \therefore \text{Number of elements}=1$
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$\displaystyle \textbf{Question 3. }\text{Let }A=\{x:x\in N,\ x\text{ is a multiple of }3\}\text{ and } \\ B=\{x:x\in N\text{ and }x\text{ is a multiple of }5\}. \text{ Write }A\cap B.$
$\displaystyle \text{Answer:}$
$\displaystyle A\cap B=\{x:x\in N,\ x\text{ is a multiple of }15\}$
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$\displaystyle \textbf{Question 4. }\text{Let }A\text{ and }B\text{ be two sets having }3\text{ and }6\text{ elements respectively.} \\ \text{Write the minimum number of elements that }A\cup B\text{ can have.}$
$\displaystyle \text{Answer:}$
$\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)$
$\displaystyle \text{For minimum value of }n(A\cup B),\ n(A\cap B)\text{ should be maximum}$
$\displaystyle \text{Maximum value of }n(A\cap B)=3$
$\displaystyle \therefore n(A\cup B)=3+6-3=6$
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$\displaystyle \textbf{Question 5. }\text{If }A=\{x\in C:x^2=1\}\text{ and }B=\{x\in C:x^4=1\},\text{ then write } \\ A-B\text{ and }B-A.$
$\displaystyle \text{Answer:}$
$\displaystyle A=\{1,-1\}$
$\displaystyle B=\{1,-1,i,-i\}$
$\displaystyle A-B=\phi$
$\displaystyle B-A=\{i,-i\}$
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$\displaystyle \textbf{Question 6. }\text{If }A\text{ and }B\text{ are two sets such that }A\subseteq B,\text{ then write }B'-A' \\ \text{ in terms of }A\text{ and }B.$
$\displaystyle \text{Answer:}$
$\displaystyle B'-A'=B'\cap A$
$\displaystyle =A-B$
$\displaystyle \text{Since }A\subseteq B,\ A-B=\phi$
$\displaystyle \therefore B'-A'=\phi$
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$\displaystyle \textbf{Question 7. }\text{Let }A\text{ and }B\text{ be two sets having }4\text{ and }7\text{ elements respectively.} \\ \text{Then write the maximum number of elements that }A\cup B\text{ can have.}$
$\displaystyle \text{Answer:}$
$\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)$
$\displaystyle \text{For maximum value of }n(A\cup B),\ n(A\cap B)=0$
$\displaystyle \therefore n(A\cup B)=4+7=11$
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$\displaystyle \textbf{Question 8. }\text{If }A=\{(x,y):y=\frac{1}{x},\ 0\neq x\in R\}\text{ and } \\ B=\{(x,y):y=-x,\ x\in R\},\text{ then write }A\cap B.$
$\displaystyle \text{Answer:}$
$\displaystyle y=\frac{1}{x}\text{ and }y=-x$
$\displaystyle \therefore \frac{1}{x}=-x$
$\displaystyle \therefore x^2=-1$
$\displaystyle \text{No real solution exists}$
$\displaystyle \therefore A\cap B=\phi$
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$\displaystyle \textbf{Question 9. }\text{If }A=\{(x,y):y=e^x,\ x\in R\}\text{ and } \\ B=\{(x,y):y=e^{-x},\ x\in R\},\text{ then write }A\cap B.$
$\displaystyle \text{Answer:}$
$\displaystyle e^x=e^{-x}$
$\displaystyle \therefore e^{2x}=1$
$\displaystyle \therefore x=0$
$\displaystyle y=e^0=1$
$\displaystyle \therefore A\cap B=\{(0,1)\}$
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$\displaystyle \textbf{Question 10. }\text{If }A\text{ and }B\text{ are two sets such that }n(A)=20,\ n(B)=25 \\ \text{ and }n(A\cup B)=40,\text{ then write }n(A\cap B).$
$\displaystyle \text{Answer:}$
$\displaystyle n(A\cup B)=n(A)+n(B)-n(A\cap B)$
$\displaystyle 40=20+25-n(A\cap B)$
$\displaystyle n(A\cap B)=5$
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$\displaystyle \textbf{Question 11. }\text{If }A\text{ and }B\text{ are two sets such that }n(A)=115,\ n(B)=326, \\ n(A-B)=47,\text{ then write }n(A\cup B).$
$\displaystyle \text{Answer:}$
$\displaystyle n(A-B)=n(A)-n(A\cap B)$
$\displaystyle 47=115-n(A\cap B)$
$\displaystyle n(A\cap B)=68$
$\displaystyle n(A\cup B)=115+326-68=373$