Counting proofs, help?

Anyway, I'm struggling like hell to understand counting proofs.

Corollary For a positive integer n, suppose that $X_1,X_2,...,X_n$ is a collection of n pairwise disjoin finite sets
(i.e. $i\not=j \Rightarrow X_i \cup X_j = \emptyset$)

Then $|X_1 \cup X_2 \cup ... \cup X_n|=|X_1|+|X_2|+...+|X_n|$

All its says is this is an easy proof by induction. I know you need to use addition principle, but then thats it.

Also, the proof of multiplication principle is like impossible to understand.

P.S. Yeah, this is so gay. I hate counting. Why do we even need to count or proof stuff about counting.

P.P.S. Its stupid as two chapters depends on me not sucking on this chapter.

disjoin finite sets
(i.e. $i\not=j \Rightarrow X_i \cup X_j = \emptyset$)

Can you prove that |X union Y| = |X| + |Y| if X and Y are disjoint?

Care to state it?

By which you mean "if I don't learn the basics I can't learn the more advanced stuff"? Yes, that is stupid.

This is maths. You thought you'd get away without counting?

$X \times Y =(\{x_1\}\times Y) \cup (\{x_2\}\times Y) \cup ... \cup(\{x_n\} \times Y)$

But now, if $g:N_m \rightarrow Y$ is a bijection, then $i \rightarrow (x_k,g(i))$ gives a bijection $N_{m} \rightarrow \{x_k\} \times Y$ so that $|\{x_k\} \times Y|=m$. It follows from the previous corollary(the one in this thread) that $|X \times Y|=nm$.

Anyway, I understand everything intill the last paragraph. I don't undestand $(x_k,g(i))$ or how the corollary that needed to be proven by induction will show the last bit.