Q: Write down the power set of the set {0}.

Let, A = {0}. The possible subsets of this set A are Φ and {0}, so the power set of the given set A is P(A) = {f, {0}}.

Q: Which of the following sets is finite?

(a) A = { x : x ε Z and x2 – 2x – 3 = 0} = {3, -1}. ∴ A is a finite set. (b) B = The set of natural numbers divisible by 2 = { 2, 4, 6, 8, 10,…}. ∴ B is an infinite set. (c) Since infinite number of lines pass through a point, C is an infinite set. (d) D = { -4, -3, -2,…}. Clearly D is an infinite set.

Q: Which of the following sets is non-empty?

A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. The set of natural numbers, denoted N, can be defined in either of two ways: N = {0, 1, 2, 3, ...} As per given option. (a) As no odd natural number is divisible by 2, the set A is empty. (b) Since no natural number satisfies the equation x + 5 = 0, ∴ B = Φ (c) Since 2 is an even prime number, i.e., C = {2}, C is not an empty set. (d) Since there is no natural number between 1 and 2, D is an empty set.

Q: Let A = {x: x ∈ N ∧ x is a multiple of 2} B = {x: x ∈ N ∧ x is a multiple of 5} C = {x: x ∈ N ∧ x is a multiple of 10} Describe the set A ∩ (B ∪ C)

B = {5, 10, 15,…} C = {10, 20, 30,…} B ∪ C = {5, 10, 15,…} ∪ {10, 20, 30,…} B ∪ C = {5, 10, 15,…} ∴ A ∩ (B ∩ C) = {2, 4, 6,…} ∩ {5, 10, 15,…} = {10, 20, 20,…} = C.

Q: If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {3, 5, 7, 9, 11} and B = {7, 8, 9, 10, 11}, Then compute (A – B)′.

A – B is a set of member which belong to A but do not belong to B ∴ A – B = {3, 5, 7, 9, 11} - {7, 8, 9, 10, 11} A – B = {3, 5} According to formula, (A − B)′ = U - (A – B) ∴ (A − B)′ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} - {3, 5} (A − B)′ = {2, 4, 6, 7, 8, 9, 10, 11}.

Q: If S and T are two sets such that S has 21 elements, T has 32 elements, and S ∩ T has 11 elements, how many elements does S ∪ T have?

Given in the question, n(S) = 21, n(T) = 32, n(S ∩ T) = 11, n(S ∪ T) = ? Using the formula, n(S ∪ T) = n(S) + n(T) – n(S ∩ T) = 21 + 32 – 11 = 42 Hence, S ∪ T has 42 elements.

Q: Let A = {1, 2}, B = {2, 3}. Evaluate A × B.

A × B = {1, 2} × {2, 3} = {(1, 2), (1, 3), (2, 2), (2, 3)}.

Question 1

If P = { 1, 2, 3 } and Q = { 4 , 5 } then what is the value of P × Q?

Accepted Answer

The Cartesian product X × Y of sets X , Y is the set of all ordered pairs ( x , y ). Two ordered pairs ( x1 , y1 ), ( x2 , y2 ) . Given that P = { 1, 2, 3 } and Q = { 4 , 5 } then, P x Q = { 1, 2, 3 } x { 4 , 5 } ⇒ P x Q = { 1 } x { 4 , 5 } , { 2 } x { 4 , 5 } , { 3 } x { 4 , 5 } ⇒ P x Q = { 1 , 4 } , { 1 , 5 } , { 2 , 4 } , { 2 , 5 } , { 3 , 4 }, { 3 , 5 } ⇒ P x Q = { (1 , 4 ) , ( 1 , 5 ) , ( 2 , 4 ) , ( 2 , 5 ) , ( 3 , 4 ), ( 3 , 5) }

Question 2

If A and B have n elements in common, how many elements do A × B and B × A have in common?

Accepted Answer

NA

Question 3

If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}, find (A ∩ B) ∪ (A ∩ C).

Accepted Answer

A ∩ B = {a, b, c} ∩ {c, d, e, f} A ∩ B = { c } A ∩ C = { a, b, c } ∩ { c, d, e } A ∩ C = { c } ∴ (A ∩ B) ∪ (A ∩ C) = { c }.

Question 4

If U = {a, b, c, d, e, f}, A = {a, b, c}, find (U ∪ A)′.

Accepted Answer

U ∪ A = {a, b, c, d, e, f} ∪ {a, b, c} = {a, b, c, d, e, f} = U (U ∪ A)′ = Φ.

Question 5

Let A = {1, 3, 5} and B = {x: x is an odd natural number < 6}. Which of the following is false?

Accepted Answer

NA

Question 6

Let A = {ϕ, {ϕ}, 1, {1, ϕ}, 7}. Which of the following is false?

Accepted Answer

NA

Question 7

Write down the power set of the set {0}.

Accepted Answer

Let, A = {0}. The possible subsets of this set A are Φ and {0}, so the power set of the given set A is P(A) = {f, {0}}.

Question 8

For which of the following cases A and B are equivalent?

Accepted Answer

NA

Question 9

Find the cardinal number of the following set {x: x is a natural number ≤ 30 and is divisible by 7 or 11}

Accepted Answer

NA

Question 10

Which of the following sets is empty?

Accepted Answer

NA

Question 11

Which of the following sets is finite?

Accepted Answer

NA

Question 12

Which of the following sets is finite?

Accepted Answer

(a) A = { x : x ε Z and x2 – 2x – 3 = 0} = {3, -1}. ∴ A is a finite set. (b) B = The set of natural numbers divisible by 2 = { 2, 4, 6, 8, 10,…}. ∴ B is an infinite set. (c) Since infinite number of lines pass through a point, C is an infinite set. (d) D = { -4, -3, -2,…}. Clearly D is an infinite set.

Question 13

Which of the following sets is non-empty?

Accepted Answer

A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. The set of natural numbers, denoted N, can be defined in either of two ways: N = {0, 1, 2, 3, ...} As per given option. (a) As no odd natural number is divisible by 2, the set A is empty. (b) Since no natural number satisfies the equation x + 5 = 0, ∴ B = Φ (c) Since 2 is an even prime number, i.e., C = {2}, C is not an empty set. (d) Since there is no natural number between 1 and 2, D is an empty set.

Question 14

Write down the power set of C = {1, {2}}.

Accepted Answer

NA

Question 15

Which of the following statements is true?

Accepted Answer

NA

Question 16

Let A = {x: x ∈ N ∧ x is a multiple of 2} B = {x: x ∈ N ∧ x is a multiple of 5} C = {x: x ∈ N ∧ x is a multiple of 10} Describe the set A ∩ (B ∪ C)

Accepted Answer

B = {5, 10, 15,…} C = {10, 20, 30,…} B ∪ C = {5, 10, 15,…} ∪ {10, 20, 30,…} B ∪ C = {5, 10, 15,…} ∴ A ∩ (B ∩ C) = {2, 4, 6,…} ∩ {5, 10, 15,…} = {10, 20, 20,…} = C.

Question 17

If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {3, 5, 7, 9, 11} and B = {7, 8, 9, 10, 11}, Then compute (A – B)′.

Accepted Answer

A – B is a set of member which belong to A but do not belong to B ∴ A – B = {3, 5, 7, 9, 11} - {7, 8, 9, 10, 11} A – B = {3, 5} According to formula, (A − B)′ = U - (A – B) ∴ (A − B)′ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} - {3, 5} (A − B)′ = {2, 4, 6, 7, 8, 9, 10, 11}.

Question 18

If S and T are two sets such that S has 21 elements, T has 32 elements, and S ∩ T has 11 elements, how many elements does S ∪ T have?

Accepted Answer

Given in the question, n(S) = 21, n(T) = 32, n(S ∩ T) = 11, n(S ∪ T) = ? Using the formula, n(S ∪ T) = n(S) + n(T) – n(S ∩ T) = 21 + 32 – 11 = 42 Hence, S ∪ T has 42 elements.

Question 19

Let A = {1, 2}, B = {2, 3}. Evaluate A × B.

Accepted Answer

A × B = {1, 2} × {2, 3} = {(1, 2), (1, 3), (2, 2), (2, 3)}.

Question 20

The Cartesian product A × A has 9 elements and two of them are (-1, 0) and (0, 1). Find the set A?

Accepted Answer

NA

Sets and Functions Questions