NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.3 - Sets

Question:1 Make correct statements by filling in the symbols $\subset$ or $\not\subset$ in the blank spaces :

(i) { 2, 3, 4 } _____ { 1, 2, 3, 4,5 }

(ii) { a, b, c }______ { b, c, d }

(iii) {x : x is a student of Class XI of your school}______{x : x student of your school}

(iv) {x : x is a circle in the plane} ______{x : x is a circle in the same plane with radius 1 unit}

(v) {x : x is a triangle in a plane} ______ {x : x is a rectangle in the plane}

(vi) {x : x is an equilateral triangle in a plane} ______{x : x is a triangle in the same plane}

(vii) {x : x is an even natural number} _____ {x : x is an integer}

Answer:

A set A is said to be a subset of a set B if every element of A is also an element of B.

(i). All elements {2,3,4} are also elements of {1,2,3,4,5} .

So, {2,3,4} $\subset$ {1,2,3,4,5}.

(ii).All elements { a, b, c } are not elements of{ b, c, d }.

Hence,{ a, b, c } $\not\subset${ b, c, d }.

(iii) Students of class XI are also students of your school.

Hence,{x : x is a student of Class XI of your school} $\subset$ {x : x student of your school}

(iv). Here, {x : x is a circle in the plane} $\not\subset$ {x : x is a circle in the same plane with radius 1 unit} : since a circle in the plane can have any radius

(v). Triangles and rectangles are two different shapes.

Hence,{x : x is a triangle in a plane} $\not\subset$ {x : x is a rectangle in the plane}

(vi). Equilateral triangles are part of all types of triangles.

So,{x : x is an equilateral triangle in a plane} $\subset$ {x : x is a triangle in the same plane}

(vii).Even natural numbers are part of all integers.

Hence, {x : x is an even natural number} $\subset$ {x : x is an integer}

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 divides 36}

Answer:

(i) All elements of { a, b } lie in { b, c, a }.So,{ a, b } $\subset${ b, c, a }.

Hence,it is false.

(ii) All elements of { a, e } lie in {a,e,i,o,u}.

Hence,the statements given is true.

(iii) All elements of { 1, 2, 3 } are not present in { 1, 3, 5 }.

Hence,statement given is false.

(iv) Element of { a } lie in { a, b, c }.

Hence,the statement is true.

(v). { a } $\subset$ { a, b, c }

So,the statement is false.

(vi) All elements {2,4,} lies in {1,2,3,4,6,9,12,18,36}.

Hence,the statement is true.

Question:3(i) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{3, 4} $\subset$ A

Answer:

3 $\in$ {3,4} but 3$\notin$ {1,2,{3,4},5}.

SO, {3, 4} $\not \subset$ A

Hence,the statement is incorrect.

Question:3(ii) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{3, 4} $\in$ A

Answer:

{3, 4} is element of A.

So, {3, 4} $\in$ A.

Hence,the statement is correct.

Question:3(iii) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{{3, 4}} $\subset$ A

Answer:

Here,

{ 3, 4 }$\in$ { 1, 2, { 3, 4 }, 5 }

and { 3, 4 } $\in$ {{3, 4}}

So, {{3, 4}} $\subset$ A.

Hence,the statement is correct.

Question:3(iv) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

1 $\in$ A

Answer:

Given, 1 is element of { 1, 2, { 3, 4 }, 5 }.

So,1 $\in$ A.

Hence,statement is correct.

Question:3(v) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

1 $\subset$ A

Answer:

Here, 1 is element of set A = { 1, 2, { 3, 4 }, 5 }.So,elements of set A cannot be subset of set A.

1 $\not\subset$ { 1, 2, { 3, 4 }, 5 }.

Hence,the statement given is incorrect.

Question:3(vi) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{1,2,5}$\subset$ A

Answer:

All elements of {1,2,5} are present in { 1, 2, { 3, 4 }, 5 }.

So, {1,2,5}$\subset$ { 1, 2, { 3, 4 }, 5 }.

Hence,the statement given is correct.

Question:3(vii) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{1,2,5} $\in$ A

Answer:

Here,{1,2,5} is not an element of { 1, 2, { 3, 4 }, 5 }.

So,{1,2,5} $\notin$A .

Hence, statement is incorrect.

Question:3(viii) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

{1,2,3} $\subset$ A

Answer:

Here, 3$\in$ {1,2,3}

but 3 $\notin$ { 1, 2, { 3, 4 }, 5 }.

So, {1,2,3} $\not \subset$ A

Hence,the given statement is incorrect.

Question:3(ix) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

$\phi \in A$

Answer:

$\phi$ is not an element of { 1, 2, { 3, 4 }, 5 }.

So,$\phi \notin A$.

Hence,the above statement is incorrect.

Question:3(x) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

$\phi \subset A$

Answer:

$\phi$ is subset of all sets.

Hence,the above statement is correct.

Question:3(xi) Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?

$\left \{ \phi \right \} \subset A$

Answer:

$\phi$ $\subset$ { 1, 2, { 3, 4 }, 5 }. but $\phi$ is not an element of { 1, 2, { 3, 4 }, 5 }.

$\left \{ \phi \right \} \not\subset A$

Hence, the above statement is incorrect.

Question:4(i) Write down all the subsets of the following sets

{a}

Answer:

Subsets of $\left \{ a \right \} = \phi \, and \left \{ a \right \}$.

Question:4(ii) Write down all the subsets of the following sets:

{a, b}

Answer:

Subsets of $\left \{ a,b \right \}\ are \ \phi , \left \{ a \right \},\left \{ b \right \} and \left \{ a,b \right \}$. Thus the given set has 4 subsets

Question:4(iii) Write down all the subsets of the following sets:

{1,2,3}

Answer:

Subsets of

$\left \{ 1,2,3 \right \} = \left \{ 1 \right \},\left \{ 2 \right \},\left \{ 3 \right \},\phi ,\left \{ 1,2 \right \},\left \{ 2,3 \right \},\left \{ 3,1 \right \},\left \{ 1,2,3 \right \}$

Question:4(iv) Write down all the subsets of the following sets:

$\phi$

Answer:

Subset of $\phi$ is $\phi$ only.

The subset of a null set is null set itself

Question:5 Write the following as intervals :

(i) {x : x $\in$R, – 4 $<$ x $\leq$ 6}

(ii) {x : x $\in$ R, – 12 $<$x $<$–10}

(iii) {x : x $\in$ R, 0 $\leq$ x $<$ 7}

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

Answer:

The following can be written in interval as :

(i) {x : x $\in$R, – 4 $<$ x $\leq$ 6} $= \left ( -4 ,6 \right ]$

(ii) {x : x $\in$ R, – 12 $<$x $<$–10} $= \left ( -12,-10 \right )$

(iii) {x : x $\in$ R, 0 $\leq$ x $<$ 7} $= [ 0,7)$

(iv) (iv) {x : x $\in$ R, 3 $\leq$ x $\leq$ 4} $=\left [ 3,4 \right ]$

Question:6 Write the following intervals in set-builder form :

(i) (– 3, 0)

(ii) [6 , 12]

(iii) (6, 12]

(iv) [–23, 5)

Answer:

The given intervals can be written in set builder form as :

(i) (– 3, 0) $= \left \{ x:x\in R, -3< x< 0 \right \}$

(ii) [6 , 12] $= \left \{ x:x\in R, 6\leq x\leq 12\right \}$

(iii) (6, 12] $= \left \{ x:x\in R, 6< x\leq 12\right \}$

(iv) [–23, 5)$= \left \{ x:x\in R, -23 \leq x< 5\right \}$

Question:7 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) Universal set for a set of right angle triangles can be set of polygons or set of all triangles.

(ii) Universal set for a set of isosceles angle triangles can be set of polygons.

Question:8 Given the sets 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}