Exercise 1.5 (Revised) - Chapter 1 -Sets - Ncert Solutions class 11 - Maths

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Chapter 1 - Sets | NCERT Solutions for Class 11 Maths

Ex Question 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'
(ii) $\mathrm{B}$ '
(iii) (A $\cup$ C),
(iv) $(\mathrm{A} \cup \mathrm{B})^{\prime}$
(v) (A')'
(vi) $(\mathrm{B}-\mathrm{C})$,

Answer.

Given: $\mathrm{U}=\{1,2,3,4,5,6,7,8,9\}$,
$
A=\{1,2,3,4\},
$
$B=\{2,4,6,8\}$ and $C=\{3,4,5,6\}$.
(i) $\mathrm{A}^{\prime}=\mathrm{U}-\mathrm{A}=\{1,2,3,4,5,6,7,8,9\}-\{1,2,3,4\}$
$
\Rightarrow A^{\prime}=\{5,6,7,8,9\}
$
(ii) $\mathrm{B}^{\prime}=\mathrm{U}-\mathrm{B}=\{1,2,3,4,5,6,7,8,9\}-\{2,4,6,8\}$
$
\Rightarrow B^{\prime}=\{1,3,5,7,9\}
$
(iii) $(\mathrm{A} \cup \mathrm{C})^{\prime}=\mathrm{U}-(\mathrm{A} \cup \mathrm{C})$
$
\Rightarrow(A \cup C)^{\prime}=\{1,2,3,4,5,6,7,8,9\}-(\{1,2,3,4\} \cup\{3,4,5,6\})
$

$\begin{aligned}
& \Rightarrow(A \cup C)^{\prime}=\{1,2,3,4,5,6,7,8,9\}-\{1,2,3,4,5,6\} \\
& \Rightarrow(A \cup C)^{\prime}=\{7,8,9\}
\end{aligned}$

$
\begin{aligned}
& \text { (iv) }(A \cup B)^{\prime}=U-(A \cup B) \\
& \Rightarrow(A \cup B)^{\prime}=\{1,2,3,4,5,6,7,8,9\}-(\{1,2,3,4\} \cup\{2,4,6,8\}) \\
& \Rightarrow(A \cup B)^{\prime}=\{1,2,3,4,5,6,7,8,9\}-\{1,2,3,4,6,8\} \\
& \Rightarrow(A \cup B)^{\prime}=\{5,7,9\}
\end{aligned}
$
$
\text { (v) }\left(A^{\prime}\right)^{\prime}=U-A^{\prime}=U-(U-A)
$
$
\begin{aligned}
& \Rightarrow\left(A^{\prime}\right)^{\prime}=\{1,2,3,4,5,6,7,8,9\}-(\{1,2,3,4,5,6,7,8,9\}-\{1,2,3,4\}) \\
& \Rightarrow\left(A^{\prime}\right)^{\prime}=\{1,2,3,4,5,6,7,8,9\}-\{5,6,7,8,9\} \\
& \Rightarrow\left(A^{\prime}\right)^{\prime}\{1,2,3,4\} \\
& =>\left(A^{\prime}\right)^{\prime}=A
\end{aligned}
$
(vi) $(\mathrm{B}-\mathrm{C})^{\prime}=\mathrm{U}-(\mathrm{B}-\mathrm{C})$
$
\begin{aligned}
& \Rightarrow(B-C)^{\prime}=\{1,2,3,4,5,6,7,8,9\}-(\{2,4,6,8\}-\{3,4,5,6\}) \\
& \Rightarrow(B-C)^{\prime}=\{1,2,3,4,5,6,7,8,9\}-\{2,8\} \\
& \Rightarrow(B-C)^{\prime}=\{1,3,4,5,6,7,9\}
\end{aligned}
$
Ex Question 2.

If $\mathrm{U}=\{a, b, c, d, e, f, g, h\}$, find the complement of the following sets:
(i) $\mathbf{A}=\{a \cdot b \cdot c\}$

(ii) $\mathrm{B}=\{d, e, f, g\}$
(iii) $\mathrm{C}=\{a, c, e, g\}$
(iv) $\mathbf{D}=\{f, g, h, a\}$

Answer.

Given: $\mathrm{U}=\{a, b, c, d, e, f, g, h\}$
(i) $\mathrm{A}^{\prime}=\mathrm{U}-\mathrm{A}$

$
=\{a, b, c, d, e, f, g, h\}-\{a, b, c\}=\{d, e, f, g, h\}
$
(ii) $\mathrm{B}^{\prime}=\mathrm{U}-\mathrm{B}$
$
=\{a, b, c, d, e, f, g, h\}-\{d, e, f, g\}=\{a, b, c, h\}
$
(iii) $\mathrm{C}^{\prime}=\mathrm{U}-\mathrm{C}$
$
=\{a, b, c, d, e, f, g, h\}-\{a, c, e, g\}=\{b, d, f, h\}
$
(iv) $\mathrm{D}^{\prime}=\mathrm{U}-\mathrm{D}$
$
=\{a, b, c, d, e, f, g, h\}-\{f, g, h, a\}=\{b, c, d, e\}
$
Ex Question 3.

Taking the set of natural numbers as the universal set, write down the complement of the following set:
(i) $\{x: x$ is an even natural number $\}$
(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 $\}$

(vii) $\{x: x$ is a perfect cube $\}$
(viii) $\{x: x+5=8\}$
(ix) $\{x: 2 x+5=9\}$
(x) $\{x: x \geq 7\}$
(xi) $\{x: x \in \mathrm{N}$ and $2 a+1>10\}$

Answer.

Given: $\mathrm{U}=\{x: x \in \mathrm{N}\}$
(i) Let $\mathrm{A}=\{x ; x$ is an even natural number $\}$
$
\begin{aligned}
& \therefore A^{\prime}=U-A=\{x: x \in \mathrm{N}\}-\{x: x \text { is an even natural number }\} \\
& =\{x: x \text { is an odd natural number }\}
\end{aligned}
$
(ii) Let $\mathrm{A}=\{x: x$ is an odd natural number $\}$
$\therefore \mathrm{A}^{\prime}=\mathrm{U}-\mathrm{A}=\{x: x \in \mathrm{N}\}-\{x: x$ is an odd natural number $\}$
$=\{x: x$ is an even natural number $\}$
(iii) Let $\mathrm{A}=\{x: x$ is a positive multiple of 3$\}$
$\therefore \mathrm{A}^{\prime}=\mathrm{U}-\mathrm{A}=\{x: x \in \mathrm{N}\}-\{x: x$ is a positive multiple of 3$\}$
$=\{x: x \in \mathrm{N}, x: x$ is not a positive multiple of 3$\}$
(iv) Let $\mathrm{A}=\{x: x$ is a prime number $\}$
$\therefore \mathrm{A}^{\prime}=\mathrm{U}-\mathrm{A}=\{x: x \in \mathrm{N}\}-\{x: x$ is a prime number $\}$
$=\{x: x \in \mathbb{N}, x: x$ is a positive composit number and $\mathrm{x}=1\}$
(v) Let $\mathrm{A}=\{x ; x$ is a natural number divisible by 3 and 5$\}$
$\therefore A^{\prime}=\mathrm{U}-\mathrm{A}=\{x: x \in \mathrm{N}\}-\{x: x$ is a natural number divisible by 15$\}$ $=\{x: x \in \mathrm{N}, x: x$ is not divisible by 15$\}$
(vi) Let $\mathrm{A}=\{x: x$ is a perfect square $\}$
$
\begin{aligned}
& \therefore \mathrm{A}^{\prime}=\mathrm{U}-\mathrm{A}=\{x: x \in \mathrm{N}\}-\{x: x \text { is a perfect square }\} \\
& =\{x: x \in \mathrm{N}, x: x \text { is not a perfect square }\}
\end{aligned}
$
(vii) Let $\mathrm{A}=\{x: x$ is a perfect cube $\}$

$
\begin{aligned}
& \therefore A^{\prime}=\mathrm{U}-\mathrm{A}=\{x: x \in \mathrm{N}\}-\{x: x \text { is a perfect cube }\} \\
& =\{x: x \in \mathrm{N}, x: x \text { is not a perfect cube }\}
\end{aligned}
$
(viii) Let A $=\{x: x+5=8\}=\{3\}$
$
\begin{aligned}
& \therefore \mathrm{A}^{\prime}=\mathrm{U}-\mathrm{A}=\{x: x \in \mathrm{N}\}-\{3\} \\
& =\{x: x \in \mathrm{N}, x \neq 3\}
\end{aligned}
$
(ix) Let $\mathrm{A}=\{x: 2 x+5=9\}=\{2\}$
$
\begin{aligned}
& \therefore \mathrm{A}^{\prime}=\mathrm{U}-\mathrm{A}=\{x: x \in \mathrm{N}\}-\{2\} \\
& =\{x: x \in \mathrm{N}, x \neq 2\}
\end{aligned}
$
(x) Let $\mathrm{A}=\{x: x \geq 7\}=\{7,8,9,10, \ldots \ldots \ldots .$.
$
\begin{aligned}
& \therefore A^{\prime}=\mathrm{U}-\mathrm{A}=\{x: x \in \mathrm{N}\}-\{7,8,9,10, \ldots \ldots \ldots .\} \\
& =\{1,2,3,4,5,6\}=\{x: x \in \mathrm{N}, x<7\}
\end{aligned}
$
(xi) Let $\mathrm{A}=\{x: x \in \mathrm{N}$ and $2 a+1>10\}=\{5,6,7,8, \ldots \ldots \ldots .$.
$
\begin{aligned}
& \therefore A^{\prime}=U-A=\{x: x \in N\}-\{5,6,7,8, \ldots \ldots \ldots . .\} \\
& =\{1,2,3,4\}
\end{aligned}
$

Ex Question 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) $(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$

Answer.

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

$
\begin{aligned}
& \text { (i) } \text { L.H.S. }=(A \cup B)^{\prime}=U-(A \cup B) \\
& =\{1,2,3,4,5,6,7,8,9\}-(\{2,4,6,8\} \cup\{2,3,5,7\}) \\
& =\{1,2,3,4,5,6,7,8,9\}-\{2,3,4,5,6,7,8\}=\{1,9\} \\
& \text { R.H.S. }=A^{\prime} \cap B^{\prime}=(U-A) \cap(U-B) \\
& =(\{1,2,3,4,5,6,7,8,9\}-\{2,4,6,8\}) \cap(\{1,2,3,4,5,6,7,8,9\}-\{2,3,5,7\}) \\
& =\{1,3,5,7,9\} \cap\{1,4,6,8,9\}=\{1,9\}
\end{aligned}
$
L.H.S. $=$ R. H. S.
$
\Rightarrow(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}
$
$
\begin{aligned}
& \text { (ii) L.H.S. }=(A \cap B)^{\prime} \\
& =U-(A \cap B) \\
& =\{1,2,3,4,5,6,7,8,9\}-(\{2,4,6,8\} \cap\{2,3,5,7\}) \\
& =\{1,2,3,4,5,6,7,8,9\}-\{2\} \\
& =\{1,3,4,5,6,7,8,9\} \\
& \text { R.H.S. }=A^{\prime} \cup B^{\prime}=(U-A) \cup(U-B)
\end{aligned}
$

$
\begin{aligned}
& =(\{1,2,3,4,5,6,7,8,9\}-\{2,4,6,8\}) \cup(\{1,2,3,4,5,6,7,8,9\}-\{2,3,5,7\}) \\
& =\{1,3,5,7,9\} \cup\{1,4,6,8,9\} \\
& =\{1,3,4,5,6,7,8,9\}
\end{aligned}
$
L.H.S. = R. H. S.
$
\Rightarrow(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}
$
Ex Question 5.

Draw appropriate Venn diagrams for each of the following:

(i) $(A \cup B)^{\prime}$
(ii) $\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}$
(iii) $(\mathrm{A} \cap \mathrm{B})^{\prime}$
(iv) $\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}$

Answer.

(i) In the diagrams, shaded portion represents $(\mathrm{A} \cup \mathrm{B})^{\prime}$

$\text { (ii) In the diagrams, shaded portion represents } \mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}$

$\text { (iii) In the diagrams, shaded portion represents }(\mathrm{A} \cap \mathrm{B})^{\prime}$

$\text { (iv) In the diagrams, shaded portion represents } A^{\prime} \cup B^{\prime}$

Ex Question 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 $\mathrm{A}^{\prime}$ ?

Answer.

Given: $\mathrm{U}=\{x: x$ is a triangle $\}$
$A=\left\{x: x\right.$ is a triangle and has at least one angle different from $\left.60^{\circ}\right\}$
$\therefore A^{\prime}=U-A=\left\{x: x\right.$ is a triangle and has all angles equal to $\left.60^{\circ}\right\}$
$=$ Set of all equilateral triangles
Ex Question 7.

Fill in the blanks to make each of the following a true statement:
(i) $A \cup A^{\prime}=$ $\qquad$
(ii) $\phi^{\prime} A=$ $\qquad$
(iii) $A \cap A^{\prime}=$ $\qquad$
(iv) $U^{\prime} \cap A=$ $\qquad$
Answer.

(i) $A \cup A^{\prime}=U$
(ii) $\phi \cap \mathrm{A}=\mathrm{U} \cap \mathrm{A}=\mathrm{A}$
(iii) $A \cap A^{\prime}=\phi$
(iv) $U^{\prime} \cap A=\phi^{\prime} \cap A=\phi$