The exercise goes like this:
In a class, there are 32 people. They learn different languages. 25 people learn English, 10 people learn French. We also know that 7 people learn neither of the two. How many people learn both languages?
This is not a hard exercise, I would simply draw a Venn-diagram like this:
And then I can easily find an equation to solve, knowing that there are 32 people in total: $$ (25-x)+(x)+(10-x)+(7)=32 \\ 42-x=32 \\ x=10 $$
That's fine; As I said, this is an easy exercise. My question is, how would I be able to solve such a problem differently? For example, by only using set theory identities, or something, starting with:
$$ |\mathbb{U}| = 32 \; \land |E| = 25 \; \land |F| = 10 \; \land | \overline{E \cup F} | = 7 \\ | E \cap F|=? $$