Page 79 - ICSE Math 8

(c) If A ⊂ B, then A ∪ B = B.
(d) A ⊆ (A ∪ B) and B ⊆ A ∪ B
(e) The union of a set and an empty set is the set itself, i.e., A ∪ φ = A.
(f) The union of a set with itself gives the set, i.e., A ∪ A = A.
(g) The union of a set and a universal set is the universal set, i.e., A ∪ U = U.

Maths Info

The operations of intersection, union and difference of sets correspond to the logical operations “and”, “or” and “not”
respectively of Boolean Algebra. Boolean algebra is used in the design of digital circuits of calculators and computers.

Intersection of sets
Let A and B be any two sets. The intersection of sets A and B is defined as the set which contains the
elements common to both A and B. It is denoted by A ∩ B (read as A intersection B). For example, let
A = {a, e, i, o, u} and B = {c, d, e, f, i}, then A ∩ B = {e, i}. Here, the elements e and i belong to both the
sets A and B.

Points to remember
• For any two sets A and B, we have n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
• If A and B are disjoint sets then, A ∩ B = φ, so, n(A ∩ B) = 0 and n(A ∪ B) = n(A) + n(B).
Example 9: Let P = {x : x is an even number between 3 and 15}, and Q be the set of first six multiples of 3.
Find P ∩ Q.
Solution: Sets P and Q in roster form are given by: P = {4, 6, 8, 10, 12, 14}, Q = {3, 6, 9, 12, 15, 18}
The common elements of P and Q are 6 and 12.
\ P ∩ Q = {6, 12}
Example 10: Using set theory, find the HCF of 12 and 20.
Solution: Let A be the set of factors of 12 and B be the set of factors of 20.
\ A = {1, 2, 3, 4, 6, 12} and B = {1, 2, 4, 5, 10, 20}
The set of common factors is A ∩ B given by {1, 2, 4} and the highest element of
A ∩ B is 4. So, HCF of 12 and 20 is 4.
Example 11: Using set theory, prove that 27 and 35 are co-prime numbers.
Solution: Let A be the set of factors of 27 and B be the set of factors of 35.
\ A = {1, 3, 9, 27} and B = {1, 5, 7, 35}
The set of common factors is A ∩ B given by {1}.
As, A and B have no common element except 1, therefore 27 and 35 are co-prime numbers.

Properties of intersection of sets
(a) Intersection of sets is commutative, i.e., for any two sets A and B, A ∩ B = B ∩ A.
(b) Intersection of sets is associative, i.e., for any three sets A, B and C, A ∩ (B ∩ C) = (A ∩ B) ∩ C.
(c) The intersection of a set with an empty set is the empty set, i.e., A ∩ φ = φ.
(d) The intersection of a set with a universal set is the set itself, i.e., A ∩ U = A.
(e) The intersection of a set with itself is the set itself, i.e., A ∩ A = A.
(f) Distributive property
(i) The union is distributive over the intersection of two sets, i.e.,
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(ii) The intersection is distributive over the union of two sets, i.e.,
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

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