Chapter 1 Sets EXERCISE 1.5
EXERCISE 1.5
1. Let $U=\{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): $B^{\prime}$
(iii): $(A \cup C)^{\prime}$
(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): $A^{\prime}=U-(A\cup C)=\{5,6,7,8,9\}$
(ii): $B^{\prime}=U-B=\{1,3,5,7,9\}$
(iii): $A \cup C=\{1,2,3,4,5,6\}$
$\therefore \ \ (A \cup C)^{\prime}=U-(A\cup C)=\{7,8,9\}$
(iv): $A \cup B=\{1,2,3,4,6,8\}$
$(A \cup B)^{\prime}=U-(A\cup B)=\{5,7,9\}$
(v): $(A^{\prime})^{\prime}=A=\{1,2,3,4\}$
(vi): $B-C=\{2,8\}$
$\therefore \ \ (B-C)^{\prime}=U-(B-C)=\{1,3,4,5,6,7,9\}$
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): $D=\{f, g, h, a\}$
Show Answer
Answer :
$U=\{a, b, c, d, e, f, g, h\}$
(i): $A=\{a, b, c\}$
$A^{\prime}=U-A=\{d, e, f, g, h\} $
(ii): $B=\{d, e, f, g\}$
$\therefore \ \ B^{\prime}=U-B=\{a, b, c, h\}$
(iii): $C=\{a, c, e, g\}$
$\therefore \ \ C^{\prime}=U-C=\{b, d, f, h\}$
(iv): $D=\{f, g, h, a\}$
$\therefore \ \ D^{\prime}=U-D=\{b, c, d, e\}$
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 $2 x+1>10\}$
Show Answer
Answer :
$U=N$ : Set of natural numbers.
We know tha complement of a set A is given by $A’=U-A$
(i): $\{x: x$ is an even natural number $\}^{\prime}=\{x: x$ 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): $\{x: x \ {\geq 7}^{\prime}=\{x: x \in N$ 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): $ \ (A \cup B)^{\prime}=\{2,3,4,5,6,7,8\}^{\prime}=\{1,9\} $
$\qquad A^{\prime} \cap B^{\prime}=\{1,3,5,7,9\} \cap(1,4,6,8,9)=\{1,9\}$
$\therefore \ \ (A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}$
(ii): $ \ (A \cap B)^{\prime}=\{2\}^{\prime}=\{1,3,4,5,6,7,8,9\} $
$\qquad 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}$
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 $U$ 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 :
$\mathrm{U}=$ Set of all triangles in a plane
$A=$ Set of all triangles with at-least one angle different from $60^{\circ}$
$\mathrm{A}^{\prime}=$ Set of all triangles with no angle different from $60^{\circ}$ $=$ Set of all triangles with all three angles $60^{\circ}$
(As, in an equilateral triangle, all angles are $60^{\circ}$ ) $=$ 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): $U^{\prime} \cap A=\ldots$
Show Answer
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$
$\therefore \ \ U^\prime \cap A=\Phi$
BETA
