Answer to Question #86903 in Algebra for sonali mansingh

Question

Give a direct proof, as well as a proof by contradiction, of the following statement:
‘ B A ∩ B ⊆ A ∪ for any two sets A and B .’

Answer 1

Direct proof:

Let 𝑥 ∈ 𝐴 ∩ 𝐵 ⇒ 𝑥 ∈ 𝐴 ⇒ 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵 ⇒ 𝑥 ∈ 𝐴 ∪ 𝐵 .

Therefore 𝐴 ∩ 𝐵 ⊆ 𝐴 ∪ 𝐵.

Proof by contradiction:

Suppose 𝐴 ∩ 𝐵 ⊄ 𝐴 ∪ 𝐵.

Then an element 𝑥 ∈ 𝐴 ∩ 𝐵 exists, such that 𝑥 ∉ 𝐴 ∪ 𝐵. Thus, from 𝑥 ∈ 𝐴 ∩ 𝐵 there is 𝑥 that belongs to both 𝐴 and 𝐵 (𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵). From 𝑥 ∉ 𝐴 ∪ 𝐵 there is 𝑥 that belongs to neither A nor B (𝑥 ∉ 𝐴 and 𝑥 ∉ 𝐵). It contradicts the former implication. The assumption 𝐴 ∩ 𝐵 ⊄ 𝐴 ∪ 𝐵 is false. Therefore, A∩B ⊆ A∪B.