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 $2 x+1>10}$
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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
