Let f:A----->B, D subset of A, and E subset of B. Prove that?

Question

A-f^-1(E) subset of f^-1 (B-E)

Answer 1

Given that f:A→B, D⊆A, E⊆B, we wish to prove A\f⁻¹(E) ⊆ f⁻¹(B\E)

Suppose x∈A\f⁻¹(E). Then x∉f⁻¹(E), so f(x)∈E. But f(x)∈B, so this means that f(x)∈B\E, and thus by definition x∈f⁻¹(B\E). Since every element of A\f⁻¹(E) is in f⁻¹(B\E), it follows that A\f⁻¹(E)⊆f⁻¹(B\E). Q.E.D.

Answer 2

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