Prove that AB A if and only if ABSolutionProof First assume

Prove that AB = A if and only if AB.

Solution

Proof.

First assume that A B. If x A B, then x A and x B by definition, so in particular x A. This proves

A B A. Now if x A, then by assumption x B, too, so x A B. This proves A A B. Together this implies A = A B. Now assume that A B = A. If x A, then by assumption x A B, so x A and x B. In particular, x B. This proves A B.