Question

.22.if AUB=AUB, prove that A = B?​

Answer 1

Step-by-step explanation:

Prove: (A∪B=A∩B)⟹(A=B)

Suppose x∈A, then x∈A∪B. So then x∈A∩B. Thus x∈B and thus A⊆B

Suppose x∈B, then x∈A∪B. So then x∈A∩B. Thus x∈A and thus B⊆A

Therefore, A=B

Is this correct? Another way to prove this? tips for a better proof writing?

Step-by-step explanation:

An other way

A⊂A∪B=A∩B⊂A

and

B⊂A∪B=A∩B⊂B.

Therefore

A=A∪B=A∩B=B.A simple proof by contradiction, A≠B, then either:

∃x:x∈A∧x∉B

∃x:x∈B∧x∉A

Let's look at 1 (2 is symmetrical). From 1 ⇒x∉A∩B which means (from A∩B=A∪B assumption) x∉A∪B which is a contradiction (with A∪B={x:x∈A∨x∈B}).