Chapter 1 Sets EXERCISE 1.5
EXERCISE 1.5
1. Let $\cup=\{1,2,3,4,5,6,7,8,9\}, A=\{1,2,3,4\}, B=\{2,4,6,8\}$ and $C=\{3,4,5,6\}$. Find (i) $A^{\prime}$ ( ii) $B^{\prime}$ (iiii) $(A \cup C)^{\prime}$ (iv) $(A \cup B)^{\prime}$ (v) $(A^{\prime})^{\prime}$ (vi) $(B-C)^{\prime}$
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Answer :
$U=\{1,2,3,4,5,6,7,8,9\}$
$A=\{1,2,3,4\}$
$B=\{2,4,6,8\}$
$C=\{3,4,5,6\}$
(i) $A^{\prime}=\{5,6,7,8,9\}$
(ii) $B^{\prime}=\{1,3,5,7,9\}$
(iii) $A \cup C=\{1,2,3,4,5,6\}$
$\therefore(A \cup C)^{\prime}=\{7,8,9\}$
(iv) $A \cup B=\{1,2,3,4,6,8\}$
$(A \cup B)^{\prime}=\{5,7,9\}$
(v) $(A^{\prime})^{\prime}=A=\{1,2,3,4\}$
(vi) $B-C=\{2,8\}$
$\therefore(B-C)^{\prime}=\{1,3,4,5,6,7,9\}$
2. If $U=\{a, b, c, d, e, f, g, h\}$, find the complements of the following sets :
(i) $A=\{a, b, c\}$
(ii) $B=\{d, e, f, g\}$
(iii) $C=\{a, c, e, g\}$
(iv) $D=\{f, g, h, a\}$
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Answer :
$U=\{a, b, c, d, e, f, g, h\}$
(i) $A=\{a, b, c\}$
$A^{\prime}=\{d, e, f, g, h\} $
(ii) $B=\{d, e, f, g\}$
$\therefore B^{\prime}=\{a, b, c, h\}$
(iii) $C=\{a, c, e, g\}$
$\therefore C^{\prime}=\{b, d, f, h\}$
(iv) $D=\{f, g, h, a\}$
$\therefore D^{\prime}=\{b, c, d, e\}$
3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) $\{x: x$ is an even natural number $\} \quad$
(ii) $\{x: x$ is an odd natural number $\}$
(iii) $\{x: x$ is a positive multiple of 3 $\}$
(iv) $\{x: x$ is a prime number $\}$
(v) $\{x: x$ is a natural number divisible by 3 and 5 $\}$
(vi) $\{x: x$ is a perfect square $\} \quad$
(vii) $\{x: x$ is a perfect cube $\}$
(viii) $\{x: x+5=8\}$
(ix) $\{x: 2 x+5=9\}$
(x) $\{x: x \geq 7\}$
(xi) $\{x: x \in N$ and $2 x+1>10\}$
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Answer :
$U=N$ : Set of natural numbers
(i) $\{x: x$ is an even natural number $\}=\{x: x$ is an odd natural number $\}$
(ii) $\{x: x \text{ is an odd natural number }\}^{\prime}=\{x: x$ is an even natural number $\}$
(iii) $\{x: x \text{ is a positive multiple of } 3\}^{\prime}=\ \{x: x \in N$ and $x$ is not a multiple of 3 $ \} $
(iv) $\{x: x \text{ is a prime number }\}^{\prime}=\{x: x$ is a positive composite number and $x=1\}$
(v) $\{x: x \text{ is a natural number divisible by } 3 \text{ and } 5\}^{\prime}=\{x: x$ is a natural number that is not divisible by 3 or 5 $ \}$
(vi) $\{x: x \text{ is a perfect square }\}^{\prime}=\{x: x \in N$ and $x$ is not a perfect square $\}$
(vii) $\{x: x \text{ is a perfect cube }\}^{\prime}=\{x: x \in N$ and $x$ is not a perfect cube $\}$
(viii) $\{x: x+5=8\}^{\prime}=\{x: x \in N$ and $x \neq 3\}$
(ix) $\{x: 2 x+5=9\}^{\prime}=\{x: x \in N$ and $x \neq 2\}$
(x) $\{x: x \underset{\large-}{>} 7\}^{\prime}=\{x: x \in N$ and $x<7\}$
(xi) $\{x: x \in N \text{ and } 2 x+1>10\}^{\prime}=\{x: x \in N$ and $x \leq 9 / 2\}$
4. If $U=\{1,2,3,4,5,6,7,8,9\}, A=\{2,4,6,8\}$ and $B=\{2,3,5,7\}$. Verify that
(i) $(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$
(ii) $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$
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Answer :
$U=\{1,2,3,4,5,6,7,8,9\}$
$A=\{2,4,6,8\}, B=\{2,3,5,7\}$
(i)
$ \begin{aligned} & (A \cup B)^{\prime}=\{2,3,4,5,6,7,8\}^{\prime}=\{1,9\} \\ & A^{\prime} \cap B^{\prime}=\{1,3,5,7,9\} \cap(1,4,6,8,9)=\{1,9\} \\ & \therefore(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} \end{aligned} $
(ii)
$ \begin{aligned} & (A \cap B)^{\prime}=\{2\}^{\prime}=\{1,3,4,5,6,7,8,9\} \\ & A^{\prime} \cup B^{\prime}=\{1,3,5,7,9\} \cup\{1,4,6,8,9\}=\{1,3,4,5,6,7,8,9\} \\ & \therefore(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime} \end{aligned} $
5. Draw appropriate Venn diagram for each of the following :
(i) $(A \cup B)^{\prime}$,
(ii) $A^{\prime} \cap B^{\prime}$,
(iii) $(A \cap B)^{\prime}$,
(iv) $A^{\prime} \cup B^{\prime}$
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Answer :
(ii) $A^{\prime} \cap B^{\prime}$
(iii) $(A \cap B)^{\prime}$
(iv) $A^{\prime} \cup B^{\prime}$
6. Let $\cup$ be the set of all triangles in a plane. If $A$ is the set of all triangles with at least one angle different from $60^{\circ}$, what is $A^{\prime}$ ?
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Answer :
$A^{\prime}$ is the set of all equilateral triangles.
7. Fill in the blanks to make each of the following a true statement :
(i) $A \cup A^{\prime}=\ldots$
(ii) $\phi^{\prime} \cap A=\ldots$
(iii) $A \cap A^{\prime}=$
(iv) $U^{\prime} \cap A=\ldots$
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Answer :
(i) $A \cup A^{\prime}=U$
(ii) $\Phi^{\prime } \cap A=U \cap A=A$
$\therefore \Phi^{\prime }\cap A=A$
(iii) $A \cap A^\prime=\Phi$
(iv) $ U^\prime \cap A=\Phi$
$\therefore U^\prime \cap A=\Phi$
