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this is my first post on this forum. I want to learn advanced/pure mathematics basically just because I find it really interesting and challenging and I have started to learn about proofs. I'm currently reading Velleman's book and I have reached the part in which you actually start to learn writing proofs. Since Velleman only offers solution for some of the proofs I dont know whether my proofs are actually valid. I would really appreciate if someone would be willing to quickly take a look at some proofs I write and give me some feedback.

Proposition: Proove that if F is a family of sets and A [itex]\in[/itex] F, then [itex]\cap[/itex] F [itex]\subseteq[/itex] A.

Ok I'll start with my scratch work:

Givens: A [itex]\in[/itex] F

Goal: [itex]\cap[/itex] F [itex]\subseteq[/itex] A

[itex]\cap[/itex] F [itex]\subseteq[/itex] A is equivalent to [itex]\forall[/itex] x (x [itex]\in[/itex] [itex]\cap[/itex] F -> x [itex]\in[/itex] A).

Now I let x be an arbitrary element.

Question here: Does x have to be an element or a set? Because [itex]\cap[/itex]F consists only of sets right?!

Then I assume that x [itex]\in[/itex] [itex]\cap[/itex] F.

Givens: A [itex]\in[/itex] F, x [itex]\in[/itex] [itex]\cap[/itex] F

Goal: x [itex]\in[/itex] A

Now x [itex]\in[/itex] [itex]\cap[/itex] F means that [itex]\forall[/itex] A [itex]\in[/itex] F (x [itex]\in[/itex] A) for some A.

So basically that for every element ( or set of F, since F is a family of sets) x is an element of that set. Since A [itex]\in[/itex] F, x is also an element of A.

Now the formal proof:

Let x be arbitrary. Suppose that x [itex]\in[/itex] [itex]\cap[/itex] F, which means that for all sets of F, x is an element of each of those sets. Since A is one of those sets, it follows that x is an element of A. Since x was arbitrary it follows that in general if A [itex]\in[/itex] F then [itex]\cap[/itex] F [itex]\subseteq[/itex] A.

Now although I think that my scratch work was correct, I think the formal proof still sounds incorrect. Could anybody please give my some feedback?

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# Started learning proofs - need some feedback

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