Proving image of intersection?

[SithsNGiggles]

Homework Statement

Let F be a relation from X to Y and let A and B be subsets of X. Then,

F(A \cap B) \subseteq F(A) \cap F(B)

The Attempt at a Solution

Let y \in F(A \cap B). Then, \exists x \in A \cap B, so \exists x \in A and x \in B.

Then, y \in F(A) and y \in F(B), so y \in F(A) \cap F(B).

Therefore, y \in F(A \cap B) \Rightarrow y \in F(A) \cap F(B), and hence, F(A \cap B) \subseteq F(A) \cap F(B).

I'm having trouble showing that the right side is not a subset of the left. Thanks for any help.

 

[HallsofIvy]

Homework Statement

Let F be a relation from X to Y and let A and B be subsets of X. Then,

F(A \cap B) \subseteq F(A) \cap F(B)

The Attempt at a Solution

Let y \in F(A \cap B). Then, \exists x \in A \cap B, so \exists x \in A and x \in B.

\exists x \in A\cap B such that F(x)= y. You might want to say that!

Then, y \in F(A) and y \in F(B), so y \in F(A) \cap F(B).

Therefore, y \in F(A \cap B) \Rightarrow y \in F(A) \cap F(B), and hence, F(A \cap B) \subseteq F(A) \cap F(B).

I'm having trouble showing that the right side is not a subset of the left. Thanks for any help.

That's because it may not be true! That's the reason for the "\subseteq" rather than just "\subset".

 

[SithsNGiggles]

\exists x \in A\cap B such that F(x)= y. You might want to say that!

This is actually the second part of the problem I'm on. The previous one was about the image of a union being equal to the union of the images. I mentioned your suggestion in that part.

That's because it may not be true! That's the reason for the "\subseteq" rather than just "\subset".

I understand that. I have to show that it's NOT a subset of the left hand side. I just don't know how to do that using the kind of logic I used in proving the left side was a subset of the right.

 

[HallsofIvy]

And I will say again that you can't prove that- it is not true- unless you mean proper subset.