Here's my attempt:
- f(A∩B) = f({x|x∈A∧x∈B}) = {f(x)|x∈{x|x∈A∧x∈B}}
- f(A)∩f(B) = f({x|x∈A}) ∩ f({x|x∈B}) = {f(x)|x∈{x|x∈A}} ∩ {f(x)|x∈{x|x∈B}} = {x|x∈{f(x)|x∈{x|x∈A}}∧x∈{f(x)|x∈{x|x∈B}}}
And now I'm stuck. Please help.
Answer
You have
A∩B⊂A⟹f(A∩B)⊂f(A),A∩B⊂B⟹f(A∩B)⊂f(B)
so it follows that f(A∩B)⊂f(A)∩f(B).
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