Sets

Question 1

What is a set in mathematics?

Answer 1

A set is a collection of distinct objects, considered as an object in its own right.

Or

A set is a well-defined collection of objects or elements.

Question 2

How many subsets does a set with n elements have?

Answer 2

2^n

Question 3

Properties of the Operation of Union

Answer 3

(i) A ∪ B = B ∪ A (Commutative law)
(ii) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C)
(Associative law )
(iii) A ∪ φ = A (Law of identity element, φ is the identity of ∪)
(iv) A ∪ A = A (Idempotent law)
(v) U ∪ A = U (Law of U)

Question 4

Properties of Operation of Intersection

Answer 4

(i) A ∩ B = B ∩ A (Commutative law).
(ii) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law).
(iii) φ ∩ A = φ, U ∩ A = A (Law of φ and U).
(iv) A ∩ A = A (Idempotent law)
(v) A ∩ U = A

Question 5

Distributive Laws

Answer 5

(1) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law) i. e.,
∩ distributes over ∪
(2) A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) (Distributive law) i. e.,
∪ distributes over ∩

Question 6

Difference of Sets

Answer 6

(1) A - B = only A
(2) B - A = only B
(3) A Δ B=(A - B) U (B - A)= {x: x ∉ A ∩ B} (symmetric difference)

Question 7

Properties of Complement Sets

Answer 7

1. Complement laws: (i) A ∪ A′ = U (ii) A ∩ A′ = φ
2. De Morgan’s law: (i) (A ∪ B)´ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′
3. Law of double complementation : (A′)′ = A
4. Laws of empty set and universal set φ′ = U and U′ = φ

Question 8

More Results of Operations On Sets

Answer 8

If A and B are any two sets, then

(i) A - B = A ∩ B’
(ii) B - A = B ∩ A’
(iii) A - B = A ⇔ A ∩ B = φ
(iv) (A - B) U B = A U B
(v) (A - B) ∩ B = φ
(vi) A ⊆ B ⇔ B ⊆ A’
(vii) (A - B) U (B - A) = (A U B) - (A ∩ B)

Question 9

If A, Band C are any three sets -

Answer 9

(i) A - (B ∩ C) = (A - B) U (A - C)

(ii) A - (B U C) = (A - B) ∩ (A - C)

(iii) A ∩ (B - C) = (A ∩ B) - (A ∩ C)

(iv) A U (B Δ C) = (A ∩ B) Δ (A ∩ C)

Question 10

IMPORTANT RESULTS ON NUMBER OF ELEMENTS IN SETS

Answer 10

(i) n (A U B) = n(A) + n(B) - n (A 2 B)

(ii) (A - B) =n(A) - n (A ∩ B) i.e. n (A - B) +n (A ∩ B) =n(A)

(iv) n (A Δ B) = No. of elements which belong to exactly one of A or B

                  = n ((A - B) U (B - A))

                  = n (A -B) +n (B - A)    [(A - B) and (B-A) 
                                                                  are disjoint]

                  = n(A) - n (A ∩ B) +n(B) (A ∩ B)

                      = n(A) + n(B) - 2n (A ∩ B)

(v) n (A U B U C) =n(A) + (B) +n(C) - n (A ∩ B)
- n (B ∩ C) - n (A ∩ C)
+n (A ∩ B ∩ C)

(vi) Number of elements in exactly two of the sets A, B, C
= n (A ∩ B) +n (B ∩ C) +n (C ∩ A)
- 3n (A ∩ B ∩ C)

(vii) Number of elements in exactly one of the sets A, B, C

                       = n (A) +n(B) +n(C) - 2n (A ∩ B) 
                            - 2n (B ∩ C) - 2n (A ∩ C) 
                             \+ 3n (A ∩ B ∩ C) 

(viii) n (A’ U B’) =n ((A ∩ B)’) = n (U) —n (A ∩ B)

(ix) n (A’ ∩ B’) =n ((A U B)’) = n (U) —n (A U B)