let A and B be sets. Prove that if A ∪ B = B, then A ∩ B = A. Is the converse true? If so, prove it, if not, find a counter example

Question

graduate math proofs in mathematics 

Answer 1

If

AUB = A + B - A∩B

       = B + A - A∩B

       = B

Then

A - A∩B = ∅ or A = A∩B

 

Conversely, if A = A∩B ⇒ A - A∩B = ∅, then

AUB = A + B - A∩B
       = B + A - A∩B
       = B + ∅

       = B