Let f:X→Y, and A⊆Y. Show that f−1(Ac)=(f−1(A))c
I know how to prove that f−1(Ac)⊆(f−1(A))c, but stuck on proving (f−1(A))c⊆f−1(Ac). Could someone help with this step please? Thanks.
Answer
Suppose x∈(f−1(A))c. Then x∈X and x∉f−1(A). Thus f(x)∉A, and of course f(x)∈Y, so f(x)∈Ac. Therefore x∈f−1(Ac).
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