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A = {Prime numbers less than 10} and 

B = {Positive odd numbers less than 10} 

∴ A = {2, 3, 5, 7},B = {1,3,5, 7, 9} 

∴ (A ∩ B) = {2, 3, 5, 7} ∩ {1,3, 5,7, 9} = 

{3,5,7} — (1)

and (B – A) = {1, 3, 5, 7, 9} – {2,3, 5,7} = 

{1,9} —–(2)

A = {x/x = 3ⁿ, n ∈ N and n < 5}

Every odd number has an odd cube 

Odd numbers can be represented as 2n+1. 

{2n+1:n ϵ W} or 

{1,3,5,7,……}

NOTE: A set is said to be equal with another set if all elements of both the sets are equal and same. 

A = {1, 2, 3} .

B = {x ∈ R:x² – 2x + 1 = 0} 

x² – 2x + 1 = 0 

(x–1)² = 0 

∴ x = 1. 

B = {1} 

C = {1, 2, 2, 3} 

In sets we do not repeat elements hence C can be written as {1, 2, 3} 

D = {x ∈R : x³ – 6x² + 11x – 6 = 0}. 

For x = 1 

= (1)³ – 6(1)²+11(1) – 6 

= 1 – 6 +11 – 6 

= 0 

For x =2 

= (2)³ – 6(2)² + 11(2) – 6 

= 8 – 24 + 22 – 6 

= 0 

For x =3 

= (3)³ – 6(3)² + 11(3) – 6 

= 27 – 54 + 33 – 6 

= 0 

As cubic equation has three roots at max so the roots are 1, 2, 3 

∴ D = {1, 2, 3} 

Hence Set A, C and D are equal.

i) Finite 

ii) Finite 

iii) Finite

Correct option is (C) {x : x² + 1 = 0 x ∈ N}

Here we can see denominator is square of numerator +1. 

We can set builder form as

⇒ [x:\(\frac{x}{x^2+1}\),0 < x < 8,x € N]

(i) In a plane there can be infinite concentric circles. Hence it is an infinite set. 

(ii) There are just 26 letters in English Alphabets and are finite. Hence it is finite set. 

(iii) It is an infinite set because, natural numbers greater than 5 is infinite. 

(iv) It is a finite set. Natural numbers start from 1 and there are 199 numbers less than 200. Hence it is finite. 

(v) It is an infinite set. Because integers less than 5 are infinite. 

(vi) It is an infinite set. Because between two real numbers, there are infinite real numbers.

(i) There is no odd natural number divisible by 2. Therefore, given set is an empty set / nullset. 

(ii) 2 is the even prime number. 

∴ Given set = {2} ≠φ 

(iii) There is no natural number x, which is simultaneously less than 5 and greater than 7. So, given set is null set. 

(iv) We know that two parallel lines have no common point and hence the given set is null set.

Here, U = all the students of a class.

 A = Students who secured 50% or more marks in Maths. 

∴ A’= Students who secured less than 50% marks in Maths.

Note: The Empty set is the set which does not contain any element. Most of the people get confused whether {0} is an empty set or not. It is not because it contains an element 0. 

(i) All numbers ending with 0. Except 0 is divisible by 5 and are even. Hence it is not an example of empty set. 

(ii) 2 is a prime number and is even, and it is the only prime which is even. So no this not an example of the empty set. 

(iii) There is not natural number whose square is 2. So it is an example of empty set. 

(iv) Never can a number be simultaneously less than 8 and greater than 12. Hence it is an example of the empty set. 

(v) No two parallel lines can never have a common point. Hence it is an example of empty set.

Correct option is A) 27

(i) Since, all numbers ending with 0. Here, except 0 is divisible by 5 and are even natural number. Thus it is not an example of empty set.

(ii) Since, 2 is a prime number and is even, and it is the only prime which is even. Therefore this not an example of the empty set.

(iii) Here, x² – 2 = 0, x² = 2, x = ± √2 ∈ N. There is not natural number whose square is 2. Therefore it is an example of empty set.

(iv) There is no natural number less than 8 and greater than 12. Thus it is an example of the empty set.

(v) Here, no two parallel lines intersect at each other. Thus it is an example of empty set.

Correct option is  (A) Colors of the rainbow

(i) Not empty

(ii) Empty

(iii) Empty

(iv) Not empty

(v) Not empty

(vi) Empty

(vii) Not empty

Correct option is B) 41

Correct option is  D) 35

Given; A and B are two sets such that A ⊆ B. 

(i) A ∪ B = A 

(ii) A ∩ B = B

Correct option is  (A) Set of intersecting points of parallel lines.

Hints: 

v. Here, P ⊆ M 

∴ P ∪ M = M 

∴ P ∩ (P ∪ M) = P ∩ M 

= P … [∵ P ⊆M]

Correct option is (A) P

Correct option is C) A ⊆ B

Correct option is D) A ⊆ B

(i) A = {x: x is an integer and –3 < x < 7} The elements of this set are –2, –1, 0, 1, 2, 3, 4, 5, and 6 only.
Therefore, the given set can be written in roster form as A = {–2, –1, 0, 1, 2, 3, 4, 5, 6}

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

The elements of this set are 1, 2, 3, 4, and 5 only. 

Therefore, the given set can be written in roster form as B = {1, 2, 3, 4, 5}

(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8} 

The elements of this set are 17, 26, 35, 44, 53, 62, 71, and 80 only. 

Therefore, this set can be written in roster form as
C = {17, 26, 35, 44, 53, 62, 71, 80}

(iv) D = {x: x is a prime number which is a divisor of 60} 60 = 2 × 2 × 3 × 5 

The elements of this set are 2, 3, and 5 only. 

Therefore, this set can be written in roster form as D = {2, 3, 5}.

(v) E = The set of all letters in the word TRIGONOMETRY 

There are 12 letters in the word TRIGONOMETRY, out of which letters T, R, and O are repeated. 

Therefore, this set can be written in roster form as E = {T, R, I, G, O, N, M, E, Y}

(vi) F = The set of all letters in the word BETTER 

There are 6 letters in the word BETTER, out of which letters E and T are repeated. 

Therefore, this set can be written in roster form as F = {B, E, T, R}

(i) 5 ∈ A 

(ii) 8 ∉ A 

(iii) 0 ∉ A 

(iv) 4 ∈ A 

(v) 2 ∈ A 

(vi) 10 ∉ A

The given sets in set – builder form are

(i) {x : x = 3n, n ∈ N,n < 5}

(ii) {x : x = 2n,n ∈ N,n < 6}

(iii) {x : x = 5n,n ∈ N, n < 5}

(iv) {x : x is an even natural number}

(v) {x : x = n²,n ∈ N,n ≤ 10} (vi) {x : x = n²,n ∈ N}

(vii) {x : x = n/(n + 1), n ∈ N, n < 7}

Correct option is (A) A ⊂ B