NCERT Answers for Class 11 Math's Section 1

Exercise 1.3

1. Make correct statements by filling in the symbols ⊂ or ⊄ 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 at 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}

Solution:

(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 at 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}

2. Let A= {1, 2, {3, 4}, 5}. Which of the accompanying assertions are mistaken and why?

(I) {3, 4} ⊂ A

(ii) {3, 4}} ∈ A

(iii) {{3, 4}} ⊂ A

(iv) 1 ∈ A

(v) 1⊂ A

(vi) {1, 2, 5} ⊂ A

(vii) {1, 2, 5} ∈ A

(viii) {1, 2, 3} ⊂ A

(ix) Φ ∈ A

(x) Φ ⊂ A

(xi) {φ} ⊂ A

Solution:

It is given that A= {1, 2, {3, 4}, 5}

(I) {3, 4} ⊂ An is mistaken

Here 3 ∈ {3, 4}; where, 3∉A.

(ii) {3, 4} ∈A is right

{3, 4} is a component of A.

(iii) {{3, 4}} ⊂ An is right

{3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.

(iv) 1∈A is right

1 is a component of A.

(v) 1⊂ An is erroneous

A component of a set can never be a subset of itself.

(vi) {1, 2, 5} ⊂ An is right

Every component of {1, 2, 5} is likewise a component of A.

(vii) {1, 2, 5} ∈ An is erroneous

{1, 2, 5} isn't a component of A.

(viii) {1, 2, 3} ⊂ An is wrong

3 ∈ {1, 2, 3}; where, 3 ∉ A.

(ix) Φ ∈ An is wrong

Φ isn't a component of A.

(x) Φ ⊂ An is right

Φ is a subset of each and every set.

(xi) {φ} ⊂ An is mistaken

Φ∈ {Φ}; where, Φ ∈ A.

3. What number of components has P (A), if A = Φ?

Solution:

On the off chance that A will be a set with m components

n (A) = m then n [P (A)] = 2m

If A = Φ we get n (A) = 0

n [P(A)] = 20 = 1

Hence, P (A) has one component.

4. Examine whether the following statements are true or false:

(i) {a, b} ⊄ {b, c, a}

(ii) {a, e} ⊂ {x: x is a vowel in the English alphabet}

(iii) {1, 2, 3} ⊂ {1, 3, 5}

(iv) {a} ⊂ {a. b, c}

(v) {a} ∈ (a, b, c)

(vi) {x: x is an even natural number less than 6} ⊂ {x: x is a natural number which divides 36}

Solution:

(i) False.

Here each element of {a, b} is an element of {b, c, a}.

(ii) True.

We know that a, we are two vowels of the English alphabet.

(iii) False.

2 ∈ {1, 2, 3} where, 2∉ {1, 3, 5}

(iv) True.

Each element of {a} is also an element of {a, b, c}.

(v) False.

Elements of {a, b, c} are a, b, c. Hence, {a} ⊂ {a, b, c}

(vi) True.

{x: x is an even natural number less than 6} = {2, 4}

{x: x is a natural number which divides 36} = {1, 2, 3, 4, 6, 9, 12, 18, 36}

5. Write down all the subsets of the following sets:

(i) {a}

(ii) {a, b}

(iii) {1, 2, 3}

(iv) Φ

Solution:

(i) Subsets of {a} are

Φ and {a}.

(ii) Subsets of {a, b} are

Φ, {a}, {b}, and {a, b}.

(iii) Subsets of {1, 2, 3} are

Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, and {1, 2, 3}.

(iv) Only subset of Φ is Φ.

6. Compose the accompanying as spans:

(I) {x: x ∈ R, - 4 < x ≤ 6}

(ii) {x: x ∈ R, - 12 < x < - 10}

(iii) {x: x ∈ R, 0 ≤ x < 7}

(iv) {x: x ∈ R, 3 ≤ x ≤ 4}

Solution:

(I) {x: x ∈ R, - 4 < x ≤ 6} = (- 4, 6]

(ii) {x: x ∈ R, - 12 < x < - 10} = (- 12, - 10)

(iii) {x: x ∈ R, 0 ≤ x < 7} = [0, 7)

(iv) {x: x ∈ R, 3 ≤ x ≤ 4} = [3, 4]

7.What all-inclusive set (s) could you propose for every one of the accompanying?

(I) The arrangement of right triangles

(ii) The arrangement of isosceles triangles

Solution:

(I) Among the arrangement of right triangles, the general set is the arrangement of triangles or the arrangement of polygons.

(ii) Among the arrangement of isosceles triangles, the widespread set is the arrangement of triangles or the arrangement of polygons or the arrangement of two-layered figures.

8.Compose the accompanying spans in set-developer structure:

(I) (- 3, 0)

(ii) [6, 12]

(iii) (6, 12]

(iv) [-23, 5)

Solution:

(I) (- 3, 0) = {x: x ∈ R, - 3 < x < 0}

(ii) [6, 12] = {x: x ∈ R, 6 ≤ x ≤ 12}

(iii) (6, 12] = {x: x ∈ R, 6 < x ≤ 12}

(iv) [-23, 5) = {x: x ∈ R, - 23 ≤ x < 5}

9.Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the next might be considered as universals set (s) for every one of the three sets A, B and C?

(I) {0, 1, 2, 3, 4, 5, 6}

(ii) Φ

(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(iv) {1, 2, 3, 4, 5, 6, 7, 8}

Solution:

(I) We know that A ⊂ {0, 1, 2, 3, 4, 5, 6}

B ⊂ {0, 1, 2, 3, 4, 5, 6}

So, C ⊄ {0, 1, 2, 3, 4, 5, 6}

Subsequently, the set {0, 1, 2, 3, 4, 5, 6} can't be the widespread set for the sets A, B, and C.

(ii) A ⊄ Φ, B ⊄ Φ, C ⊄ Φ

Thus, Φ can't be the general set for the sets A, B, and C.

(iii) A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Subsequently, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the widespread set for the sets A, B, and C.

(iv) A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

So, C ⊄ {1, 2, 3, 4, 5, 6, 7, 8}

Subsequently, the set {1, 2, 3, 4, 5, 6, 7, 8} can't be the general set for the sets A, B, and C.

Exercise 1.4

1. Track down the association of every one of the accompanying sets of sets:

(I) X = {1, 3, 5} Y = {1, 2, 3}

(ii) A = {a, e, I, o, u} B = {a, b, c}

(iii) A = {x: x is a characteristic number and different of 3}

B = {x: x is a characteristic number under 6}

(iv) A = {x: x is a characteristic number and 1 < x ≤ 6}

B = {x: x is a characteristic number and 6 < x < 10}

(v) A = {1, 2, 3}, B = Φ

Solution:

(I) X = {1, 3, 5} Y = {1, 2, 3}

So, the association of the sets of set can be composed as

X ∪ Y= {1, 2, 3, 5}

(ii) A = {a, e, I, o, u} B = {a, b, c}

So, the association of the sets of set can be composed as

A∪ B = {a, b, c, e, I, o, u}

(iii) A = {x: x is a characteristic number and various of 3} = {3, 6, 9 …}

B = {x: x is a characteristic number under 6} = {1, 2, 3, 4, 5, 6}

So, the association of the sets of set can be composed as

A ∪ B = {1, 2, 4, 5, 3, 6, 9, 12 …}

Subsequently, A ∪ B = {x: x = 1, 2, 4, 5 or a various of 3}

(iv) A = {x: x is a characteristic number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}

B = {x: x is a characteristic number and 6 < x < 10} = {7, 8, 9}

So, the association of the sets of set can be composed as

A∪ B = {2, 3, 4, 5, 6, 7, 8, 9}

Thus, A∪ B = {x: x ∈ N and 1 < x < 10}

(v) A = {1, 2, 3}, B = Φ

So, the association of the sets of set can be composed as

A∪ B = {1, 2, 3}

2. If A and B are two sets such that A ⊂ B, then what is A ∪ B?

Solution:

If A and B are two sets such that A ⊂ B, then A ∪ B = B.

3. Find the intersection of each pair of sets:

(i) X = {1, 3, 5} Y = {1, 2, 3}

(ii) A = {a, e, i, o, u} B = {a, b, c}

(iii) A = {x: x is a natural number and multiple of 3}

B = {x: x is a natural number less than 6}

(iv) A = {x: x is a natural number and 1 < x ≤ 6}

B = {x: x is a natural number and 6 < x < 10}

(v) A = {1, 2, 3}, B = Φ

Solution:

(i) X = {1, 3, 5}, Y = {1, 2, 3}