F(A ∩ B) ⊆ F(A) ∩ F(B) laymen translation?

Question

I am suppose to prove the above statement but i have got diffculty understanding it in the first place. Could anyone help me translate it into laymen language?

Answer 1

It is saying that "the image of the intersection of $A$ and $B$ is contained in the intersection of the images of $A$ and $B$". You just have to take $y \in F(A \cap B)$ and prove that it is in $F(A) \cap F(B)$.

I'll begin it for you. Take $y \in F(A \cap B)$. Then, by definition of $F(A \cap B)$, exists $x \in A \cap B$ such that $y = F(x)$. Then...

Answer 2

From what I can understand, it's saying if you have a domain of $A \cap B$ and plug this in the the function $F$, then you get a range which is a subset of the range if you had two domains: $A$ and $B$.

For instance, let's say $F(x) = x$ and $A = \{1,2\}$ and $B = \{2,3\}$. Then the range of $F(A \cap B) = \{1,2,3\}$. The range of $F(A) = \{1,2\}$ and the range of $F(B) = \{2,3\}$. We can see that the range of $F(A) \cap F(B) = F(A \cap B)$ in this case, so it is certainly a subset.