Class 11 Maths Chapter 1 – Sets Exercise 1.5

Exercise 1.5

Q1. 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) $A'$

(ii) $B'$

(iii) $(A\cup C)'$

(iv) $(A\cup B)'$

(v) $(A')'$

(vi) $(B-C)'$

Given, $U=\{1,2,3,4,5,6,7,8,9\}$

$A=\{1,2,3,4\},\quad B=\{2,4,6,8\}$

$C=\{3,4,5,6\}$

\begin{aligned} \textbf{(i)}\ A' &= U-A \\ &= \{1,2,3,4,5,6,7,8,9\}-\{1,2,3,4\} \\ &= \{5,6,7,8,9\} \end{aligned}

\begin{aligned} \textbf{(ii)}\ B' &= U-B \\ &= \{1,2,3,4,5,6,7,8,9\}-\{2,4,6,8\} \\ &= \{1,3,5,7,9\} \end{aligned}

\begin{aligned} \textbf{(iii)}\ A\cup C &= \{1,2,3,4\}\cup\{3,4,5,6\} \\ &= \{1,2,3,4,5,6\} \end{aligned}

∴

\begin{aligned} & (A\cup C)'=U-(A\cup C) \\ & {=}\;\{1,2,3,4,5,6,7,8,9\} -\{1,2,3,4,5,6\} \\ & {=}\;\{7,8,9\} \end{aligned}

\begin{aligned} \textbf{(iv)}\ A\cup B &= \{1,2,3,4\}\cup\{2,4,6,8\} \\ &= \{1,2,3,4,6,8\} \end{aligned}

∴

\begin{aligned} & (A\cup B)'=U-(A\cup B) \\ & {=}\;\{1,2,3,4,5,6,7,8,9\} -\{1,2,3,4,6,8\} \\ & {=}\;\{5,7,9\} \end{aligned}

(v) From (i), $A'=\{5,6,7,8,9\}$ .

∴

\begin{aligned} & (A')'=U-A' \\ & {=}\;\{1,2,3,4,5,6,7,8,9\}-\{5,6,7,8,9\} \\ & {=}\;\{1,2,3,4\} \\ & {=}\;A \end{aligned}

\begin{aligned} \textbf{(vi)}\ B-C &= \{2,4,6,8\}-\{3,4,5,6\} \\ &= \{2,8\} \end{aligned}

∴

\begin{aligned} & (B-C)'=U-(B-C) \\ & {=}\;\{1,2,3,4,5,6,7,8,9\}-\{2,8\} \\ & {=}\;\{1,3,4,5,6,7,9\} \end{aligned}

Q2. 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) $D=\{f,g,h,a\}$

Sol. Given, $U=\{a,b,c,d,e,f,g,h\}$ .

\begin{aligned} \textbf{(i)}\ A' &= U-A \\ &= \{a,b,c,d,e,f,g,h\}-\{a,b,c\} \\ &= \{d,e,f,g,h\} \end{aligned}

\begin{aligned} \textbf{(ii)}\ B' &= U-B \\ &= \{a,b,c,d,e,f,g,h\}-\{d,e,f,g\} \\ &= \{a,b,c,h\} \end{aligned}

\begin{aligned} \textbf{(iii)}\ C' &= U-C \\ &= \{a,b,c,d,e,f,g,h\}-\{a,c,e,g\} \\ &= \{b,d,f,h\} \end{aligned}

\begin{aligned} \textbf{(iv)}\ D' &= U-D \\ &= \{a,b,c,d,e,f,g,h\}-\{f,g,h,a\} \\ &= \{b,c,d,e\} \end{aligned}

Q3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:

Given, the universal set $U=\mathbb N$ , the set of natural numbers.

(i) $\{x:x\text{ is an even natural number}\}$

Sol. The complement is the set of odd natural numbers.

\{x:x\text{ is an odd natural number}\}

(ii) $\{x:x\text{ is an odd natural number}\}$

Sol. The complement is the set of even natural numbers.

\{x:x\text{ is an even natural number}\}

(iii) $\{x:x\text{ is a positive multiple of 3}\}$

Sol. The complement is the set of natural numbers that are not multiples of $3$ .

\{x:x\in\mathbb N\text{ and }x\text{ is not a multiple of }3\}

(iv) $\{x:x\text{ is a prime number}\}$

Sol. A natural number that is not prime is either $1$ or a composite number.

∴ The complement is

\{x:x=1\text{ or }x\text{ is a composite number}\}

(v) $\{x:x$ is a natural number divisible by 3 and $\text{5}\}$

Sol. A natural number divisible by both $3$ and $5$ is divisible by $15$ .

∴ The complement is

\{x:x\in\mathbb N\text{ and }x\text{ is not divisible by }15\}

(vi) $\{x:x\text{ is a perfect square}\}$

Sol. The complement is

\{x:x\in\mathbb N\text{ and }x\text{ is not a perfect square}\}

(vii) $\{x:x\text{ is a perfect cube}\}$

Sol. The complement is

\{x:x\in\mathbb N\text{ and }x\text{ is not a perfect cube}\}

(viii) $\{x:x+5=8\}$

Sol. $x+5=8\Rightarrow x=3$

∴ The complement is $\{x:x\in\mathbb N,\ x\ne3\}$ .

(ix) $\{x:2x+5=9\}$

Sol. $2x+5=9\Rightarrow 2x=4\Rightarrow x=2$

∴ The complement is $\{x:x\in\mathbb N,\ x\ne2\}$ .

(x) $\{x:x\ge7\}$

Sol. The natural numbers less than $7$ are $1,2,3,4,5,6$ .

∴ The complement is

\{1,2,3,4,5,6\}

(xi) $\{x:x\in N\text{ and }2x+1>10\}$

Sol. $2x+1>10\Rightarrow 2x>9\Rightarrow x>\frac92$

∴ The complement is

\left\{x:x\in\mathbb N,\ x\le\frac92\right\}=\{1,2,3,4\}

Q4. 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) $(A\cup B)'=A'\cap B'$

(ii) $(A\cap B)'=A'\cup B'$

Sol. Given, $U=\{1,2,3,4,5,6,7,8,9\}$ , $A=\{2,4,6,8\}$ and $B=\{2,3,5,7\}$ .

Here,

\begin{array}{l} \displaystyle A'=U-A \\[0.4em] \hspace{1em} \displaystyle =\{1,2,3,4,5,6,7,8,9\}-\{2,4,6,8\} \\[0.4em] \hspace{1em} \displaystyle =\{1,3,5,7,9\} \end{array}

and

\begin{array}{l} \displaystyle B'=U-B \\[0.4em] \hspace{1em} \displaystyle =\{1,2,3,4,5,6,7,8,9\}-\{2,3,5,7\} \\[0.4em] \hspace{1em} \displaystyle =\{1,4,6,8,9\} \end{array}

(i) LHS:

\begin{aligned} A\cup B &= \{2,4,6,8\}\cup\{2,3,5,7\} \\ &= \{2,3,4,5,6,7,8\} \end{aligned}

\begin{aligned} & (A\cup B)'=U-(A\cup B) \\ & {=}\;\{1,2,3,4,5,6,7,8,9\} -\{2,3,4,5,6,7,8\} \\ & {=}\;\{1,9\}\qquad...(1) \end{aligned}

RHS:

\begin{aligned} & A'\cap B'=\{1,3,5,7,9\}\cap\{1,4,6,8,9\} \\ & {=}\;\{1,9\}\qquad...(2) \end{aligned}

From (1) and (2), $(A\cup B)'=A'\cap B'$ .

Hence verified.

(ii) LHS:

\begin{aligned} A\cap B &= \{2,4,6,8\}\cap\{2,3,5,7\} \\ &= \{2\} \end{aligned}

\begin{aligned} & (A\cap B)'=U-(A\cap B) \\ & {=}\;\{1,2,3,4,5,6,7,8,9\}-\{2\} \\ & {=}\;\{1,3,4,5,6,7,8,9\}\qquad...(3) \end{aligned}

RHS:

\begin{aligned} & A'\cup B'=\{1,3,5,7,9\}\cup\{1,4,6,8,9\} \\ & {=}\;\{1,3,4,5,6,7,8,9\}\qquad...(4) \end{aligned}

From (3) and (4), $(A\cap B)'=A'\cup B'$ .

Hence verified.

Q5. Draw appropriate Venn diagram for each of the following:

(i) $(A\cup B)'$

Sol.

(ii) $A'\cap B'$

Sol.

(iii) $(A\cap B)'$

Sol.

(iv) $A'\cup B'$

Sol.

Q6. 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 $A'$ ?

Sol. $A'$ is the set of those triangles in the plane which are not in $A$ , i.e., the triangles in which no angle is different from $60^\circ$ .

So, every angle of such a triangle is $60^\circ$ , i.e., the triangle is equilateral.

∴ $A'$ is the set of all equilateral triangles in the plane.

Q7. Fill in the blanks to make each of the following a true statement:

(i) $A\cup A'=\ldots$

(ii) $\phi'\cap A=\ldots$

(iii) $A\cap A'=\ldots$

(iv) $U'\cap A=\ldots$

(i) $A\cup A'=U$

(ii) $\phi'\cap A=U\cap A=A$

(iii) $A\cap A'=\phi$

(iv) $U'\cap A=\phi\cap A=\phi$

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