# Math Help - Set Problem

1. ## Set Problem

Let A, B and C be three non-empty subsets of a set X and f be a function from X to a set Y (i.e.,f:X→Y).Given some arbitrary D⊆X,by f(D) we denote the set {y ∈ Y |y = f (x) for some x ∈ D}. Prove the following: (a) If A ∩ B ̸= ∅, then f(A ∩ B) ⊆ f(A) ∩ f(B);
(b) f(A ∪ B) = f(A) ∪ f(B).

Been having real problems with this problem. I have just started to study martix algebra so go easy if this might be simple.

2. Originally Posted by hello123
Let A, B and C be three non-empty subsets of a set X and f be a function from X to a set Y (i.e.,f:X→Y).Given some arbitrary D⊆X,by f(D) we denote the set {y ∈ Y |y = f (x) for some x ∈ D}. Prove the following: (a) If A ∩ B ̸= ∅, then f(A ∩ B) ⊆ f(A) ∩ f(B);
(b) f(A ∪ B) = f(A) ∪ f(B).

Been having real problems with this problem. I have just started to study martix algebra so go easy if this might be simple.
What does $f\left(A\right)f\left(B\right)=f\left(AB\right)$ even mean? Cartesian product?

For teh second one merely note that if $x\in f\left(A\cup B\right)\Leftrightarrow f^{-1}(x)\in A\cup B\Leftrightarrow f^{-1}(x)\in A\text{ or }f^{-1}(x)\in B\Leftrightarrow$ $x\in f(A)\text{ or }x\in f(B)\Leftrightarrow f(x)\in\left(f(A)\cup f(B)\right)$