If a and b are two sets then prove that A-B=A if and only if a intersectionb=0

Dear Student,
Suppose A - B = A. Assume p is an arbitrary element in the intersection of A and B; then p is in A and p is in B. Since p is in A and A = A - B, we have p is not in B, contradicting our assumption. Thus there are no elements in the intersection of A and B. That is, A (intersect) B is the null set.

Conversely, we'll show the contrapositive. Suppose A - B is not equal to A. Now, since

A = (A - B) union (A intersect B),

we must have A intersect B is nonempty.

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