Exercise 1.3 (Revised) - Chapter 1 -Sets - Ncert Solutions class 11 - Maths

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Chapter 1 - Sets | NCERT Solutions for Class 11 Maths

Ex 1.3 Question 1.

$\text {Make correct statements by filling in the symbols } \subset \text { or } \not \text { in the blank spaces: }$

(i) $\{2,3,4\}$ $\qquad$ $\{1,2,3,4,5\}$
(ii) $\{a, b, c\}$ $\qquad$ $\{b, c, d\}$
(iii) $\{x: x$ is a student of class XI of your school $\}$ $\qquad$ $\{x: x$ student of your school $\}$
(iv) $\{x: x$ is a circle in the plane $\}$ $\qquad$ $\{x: x$ is a circle in the same plane with radius 1 unit\}
(v) $\{x: x$ is a triangle in plane $\}$ $\qquad$ $\{x: x$ is a rectangle in the same plane $\}$
(vi) $\{x: x$ is an equilateral triangle in a plane $\}$ $\qquad$ $\{x: x$ is a rectangle in the same plane\}
(vii) $\{x: x$ is an even natural number $\}$ $\qquad$ $\{x: x$ is an integer $\}$

Answer.

(i) $\subset$
(ii) $\Psi$
(iii) $\subset$
(iv) $\not$
(v) $Z$
(vi) $\subset$

$
\text { (vii) } \subset
$
Ex 1.3 Question 2.

Examine whether the following statements are true or false:

(i) $\{a, b\} \not \subset\{b, c, a\}$
(ii) $\{a, e\} \subset\{x: x$ is a vowel in the English alphabet $\}$
(iii) $\{1,2,3\} \subset\{1,3,5\}$
(iv) $\{a\} \subset\{a, b, c\}$
(v) $\{a\} \in\{a, b, c\}$
(vi) $\{x: x$ is an even natural number less than 6$\} \subset\{x: x$ is a natural number which divide 36$\}$

Answer.

(i) Let $\mathrm{A}=\{a, b\}$ and $\mathrm{B}=\{b, c, a\}$

Here, every element of set $\mathrm{A}$ is an element of set $\mathrm{B}$.
$
\therefore \mathrm{A} \subset \mathrm{B}
$

Therefore, statement is false.
(ii) Let $\mathrm{A}=\{a, e\}$ and $\mathrm{B}$
$=\{x: x$ is a vowel in the English alphabet $\}$
$
=\{a, e, i, o, u\}
$

Here, every element of set A is an element of set B.
$
\therefore \mathrm{A} \subset \mathrm{B}
$

Therefore, statement is true.
(iii) Let $A=\{1,2,3\}$ and $B=\{1,3,5\}$

Here, 2 A but 2 \& $\mathrm{B}$
$
\therefore \mathrm{A} \not \mathrm{B}
$

Therefore, statement is false.
(iv) Let $\mathrm{A}=\{a\}$ and $\mathrm{B}=\{a, b, c\}$

Here, every element of set A is an element of set B.
$
\therefore \mathrm{A} \subset \mathrm{B}
$

Therefore, statement is true.
(v) Let $\mathrm{A}=\{a\}$ and $\mathrm{B}=\{a, b, c\}$

Here, $\{a\} \notin B$
Therefore, statement is false.
(vi) Let $\mathrm{A}=\{x: x$ is an even natural number less than 6$\}$
$
=\{2,4\}
$

And $B=\} \subset\{x: x$ is a natural number which divide 36$\}$
$
=\{1,2,3,4,6,12,18,36]
$

Here, every element of set A is an element of set B.
$
\therefore \mathrm{A} \subset \mathrm{B}
$

Therefore, statement is true.
Ex 1.3 Question 3.

Let $A=\{1,2,\{3,4\}, 5\}$. Which of the following statements are incorrect and why:
(i) $\{3,4\} \subset \mathrm{A}$
(ii) $\{3,4\} \in A$
(iii) $\{\{3,4\}\} \subset \mathrm{A}$
(iv) $1 \in \mathrm{A}$

(v) $1 \subset \mathrm{A}$
(vi) $\{1,2,5\} \subset \mathrm{A}$
(vii) $\{1,2,5\} \in \mathrm{A}$
(viii) $\{1,2,3\} \subset \mathrm{A}$
(ix) $\phi \in \mathrm{A}$
(x) $\phi \subset \mathrm{A}$
(xi) $\{\phi\} \subset A$

Answer.

(i) $\{3,4\}$ is a member of set A.
$
\Rightarrow\{3,4\} \in \mathrm{A}
$

Therefore, $\{3,4\} \subset$ A is incorrect.
(ii) $\{3,4\}$ is a member of set A. Therefore, $\{3,4\} \in$ A is correct.
(iii) $\{3,4\}$ is a member of set $\mathrm{A}$.
$\Rightarrow\{\{3,4\}\}$ is a set.
Therefore, $\{\{3,4\}\} \subset$ A is correct
(iv) 1 is a member of set A. Therefore $1 \in$ A is correct.

(v) 1 is not a set, it is a member of set A. Therefore, $1 \subset$ A is incorrect.
(vi) 1, 2, 5 are the members of set A.
$\Rightarrow\{1,2,5\}$ is a subset of set $\mathrm{A}$.
Therefore, $\{1,2,5\} \subset$ A is correct.
(vii) 1, 2, 5 are the members of set A.
$\Rightarrow\{1,2,5\}$ is a subset of set $\mathrm{A}$.

Therefore, $\{1,2,5\} \in \mathrm{A}$ is incorrect.
(viii) 3 is not a member of set $\mathrm{A}$.
$\Rightarrow\{1,2,3\}$ is not a subset of set $A$.
Therefore, $\{1,2,3\} \subset$ A is incorrect.
(ix) $\phi$ is not a member of set A. Therefore, $\phi \in$ A is incorrect.
(x) $\phi$ is subset of all sets. Therefore, $\phi \subset$ A is correct.
(xi) $\phi$ is a subset of A and it is not an element of A. So this statement is incorrect.
Ex 1.3 Question 4.

Write down all the subsets of the following sets:
(i) $\{a\}$
(ii) $\{a, b\}$
(iii) $\{1,2,3\}$
(iv) $\emptyset$

Answer.

(i) Number of elements in given set $=1$.
Number of subsets of given set $=2^1=2$
Therefore, Subsets of given set are $\phi=\{a\}$

(ii) Number of elements in given set $=2$

Number of subsets of given set $=2^2=4$
Therefore, Subsets of given set are
$
\phi,\{a\},\{b\},\{a, b\} \text {. }
$
(iii) Number of elements in given set $=3$

Number of subsets of given set $=2^3=8$
Therefore, Subsets of given set are
$
\phi,\{1\},\{2\}=\{3\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\} \text {. }
$
(iv) Number of elements in given set $=0$

Number of subsets of given set $=2^{\circ}=1$
Therefore, Subsets of given set are $\phi$.
Ex 1.3 Question 5.

How many elements has $\mathrm{P}(\mathrm{A})$, if $\mathrm{A}=\phi$ ?

Answer.

Number of elements in set $\mathrm{A}=0$
Number of subsets of given set $=2^{\circ}=1$
Therefore, number of elements of $\mathrm{P}(\mathrm{A})$ is 1 .
Ex 1.3 Question 6.

Write the following as intervals:
(i) $\{x: x \in \mathrm{R},-4<x \leq 6\}$
(ii) $\{x: x \in R,-12<x<-10\}$
(iii) $\{x: x \in \mathbf{R}, 0 \leq x<7\}$

(iv) $\{x: x \in \mathrm{R}, 3 \leq x \leq 4\}$

Answer.

(i) Let $\mathrm{A}=\{x: x \in \mathrm{R},-4<x \leq 6\}$
It can be written in the form of interval as $(-4,6]$
(ii) Let $\mathrm{A}=\{x: x \in \mathrm{R},-12<x \leq-10\}$

It can be written in the form of interval as $(-12,-10)$
(iii) Let $\mathrm{A}=\{x ; x \in \mathrm{R}, 0 \leq x<7\}$

It can be written in the form of interval as $[0,7)$
(iv) Let $\mathrm{A}=\{x: x \in \mathrm{R}, 3 \leq x \leq 4\}$

It can be written in the form of interval as $[3,4]$
Ex 1.3 Question 7.

Write the following intervals in set-builder form:
(i) $(-3,0)$
(ii) $[6,12]$
(iii) $(6,12]$
(iv) $[-23,5)$

Answer.

(i) $\{x: x \in R,-3<x<0\}$
(ii) $\{x: x \in R, 6 \leq x \leq 12\}$
(iii) $\{x: x \in \mathrm{R}, 6<x \leq 12\}$
(iv) $\{x: x \in \mathrm{R},-12 \leq x<5\}$
Ex 1.3 Question 8.

What universal set(s) would you propose for each of the following:
(i) The set of right triangles

(ii) The set of isosceles triangles

Answer.

(i) Right triangle is a type of triangle. Therefore, the set of triangles contain all types of triangles.
$\therefore \mathrm{U}=\{x: x$ is a triangle in plane $\}$
(ii) Isosceles triangle is a type of triangle. Therefore, the set of triangles contain all types of triangles.
$\therefore \mathrm{U}=\{x: x$ is a triangle in plane $\}$

Ex 1.3 Question 9.

Given the set $A=\{1,3,5\}, B=\{2,4,6\}$ and $C=\{0,2,4,6,8\}$, which of the following may be considered as universal set(s) for all the three sets $A, B$ and $C$ :
(i) $\{0,1,2,3,4,5,6\}$
(ii) $\phi$
(iii) $\{0,1,2,3,4,5,6,7,8,9,10\}$
(iv) $\{1,2,3,4,5,6,7,8\}$

Answer.

(i) $\{0,1,2,3,4,5,6\}$ is not a universal set for $\mathrm{A}, \mathrm{B}, \mathrm{C}$ because $8 \in \mathrm{C}$ but 8 is not a member of $\{0,1,2,3,4,5,6\}$.
(ii) $\phi$ is a set which contains no element. therefore, it is not a universal set for A, B, C.
(iii) $\{0,1,2,3,4,5,6,7,8,9,10\}$ is a universal set for $\mathrm{A}, \mathrm{B}, \mathrm{C}$ because all members of $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are present in $\{0,1,2,3,4,5,6,7,8,9,10\}$.
(iv) $\{1,2,3,4,5,6,7,8\}$ is not a universal set for $\mathrm{A}, \mathrm{B}, \mathrm{C}$ because $0 \in \mathrm{C}$ but 0 is not a member of $\{1,2,3,4,5,6,7,8\}$.