Proof:A = (A ∩ B) ∪ (A ∖ B)

Proof:
Let RHS denote the right hand side ().

In order to show that , we must show both that 1. and 2. .

1. Choose an arbitrary . Clearly is either in or not. In the first case, we have that so (by definition of ). In the second case, we have that , so again . In either case, we have as desired.

2. Now we wish to show that , so we choose an arbitrary . By definition of the union, we know that either or . In the first case, we have (by definition of ), while in the second case we have by definition of . In either case we have as

desired.