X and Y are nonempty sets and f: X -> Y is a function with domain X and codomain

Question

X and Y are nonempty sets and f: X -> Y is a function with domain X and codomain Y. The direct image f(D) is denoted through f of a subset D included in X. Let A and B be any two subsets of X. Prove that:

f(A) minus f(B) is a subset of f(A minus B)

Explanation / Answer

x not in B ==> x not in A. (1) If x is in C - B, then x is in C and x is not in B. By the above note, this means that x is in C and x is not in A. Hence x is in C - A. (2) The converse is false. Let C = {1}, A = {1,2}, B = {1}. So, C - B = C - A are both empty sets (hence C - B is a subset of C - A), but A is not a subset of B. I hope that helps!

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