The intersection of two sets A and B is a set that contains elements that are in both A and B. The symbol for ‘intersection of’ is ‘n’.
A ∩ B = { \(x\) | \(x\) ∈ A and x ∈ B}
Ex. A = { letters of the word ‘fun’}
B = { letters of the word ‘son’}
Then, A ∩ B = { n }
Properties of Intersection of Sets
(i) If A is any set, then
(a) A ∩ ϕ = ϕ (b) A ∩ ξ = A
(c) A ∩ A = A (d) A ∩ A′ = ϕ
(ii) If A, B and C be any sets, then
(a) A ∩ B = B ∩ A (Commutative law)
(b) (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law)
(c) If A ⊆ B, then A ∩ B = A
(d) For any sets A and B, we have A ∩ B ⊆ A and A ∩ B ⊆ B
Venn diagrams illustrating the intersection of sets
The shaded portions in the following diagrams illustrate the intersection of the given sets.
Note : (A ∪ B)′ = A′ ∩ B ′
(A ∩ B)′ = A′ ∪ B ′
