Let f: X→Y be a map with A1, A2⊂X and B1,B2⊂Y (A) Prove f(A1∪A2)=f(A1)∪f(A2). (B) Prove...

Let f: X→Y be a map with A₁, A₂⊂X and B₁,B₂⊂Y

(A) Prove f(A₁∪A₂)=f(A₁)∪f(A₂).

(B) Prove f(A₁∩A₂)⊂f(A₁)∩f(A₂). Give an example in which equality fails.

(C) Prove f⁻¹(B₁∪B₂)=f⁻¹(B₁)∪f⁻¹(B₂), where f⁻¹(B)={x∈X: f(x)∈B}.

(D) Prove f⁻¹(B₁∩B₂)=f⁻¹(B₁)∩f⁻¹(B₂).

(E) Prove f⁻¹(Y∖B₁)=X∖f⁻¹(B₁).

(Abstract Algebra)

Expert Solution

Colby Messinger answered 2 years ago

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