Set theory venn diagram help. Homework I am new to set theory and one of our exercises is the following question:
decide on the truth or falsity of the claim that, for all sets A, B, C, D
[A ∩ B ⊆ C ∩ D] ⇒ [(A∆B) ⊇ (C∆D)].
I have drawn the following Venn diagrams
And thus, from this visual proof, I conclude that the claim is false. If someone could validate my answer that would be appreciated, but even better if someone could tell me how I could prove this using set notation that would be even better or just put me in the direction of some helpful websites.
Anyway all feedback is appreciated.
 A: You are correct that the claim is false.  Once you've decided that the claim is false, the best thing to do is find a counterexample.  To show the statement is false, all you have to do is find some way of defining sets A,B,C,D so that the claim fails.  One possible way of doing this (there may be simpler ones) is to say,
Let
$$ A = \{ 1 \}, \quad B = \{ 1 \}, \quad C = \{ 1, 2 \}, \quad D = \{ 1, 3\}. $$
From here we compute to see the left side is true,
$$ A \cap B = \{ 1 \}, \qquad C \cap D = \{ 1 \}$$
Thus indeed,
$$ A \cap B \subseteq C \cap D.$$
For the expression on the right,
$$ A \Delta B = (A \smallsetminus B) \cup (B \smallsetminus A) = \varnothing$$
And,
$$ C \Delta D = (C \smallsetminus D) \cup (D \smallsetminus C) = \{ 2,3 \}.$$
But
$$ \varnothing \nsupseteq \{ 2,3 \}$$
Hence, the expression is on the right is false.  Therefore, the statement as a whole is false.
A word on your diagram
The diagram you have drawn does not accurately represent the statement.  Once you have defined the sets $A,B,C,D$ on the left, you cannot change them in the diagram on the right.  So, the labels you have on the right don't make sense to me.  Do you mean to label things on the right $A \Delta B$ and $C \Delta D$?  You can't just label a different section with $A$, etc.
