random variables and summation

Question

First, I apologize for the quality of this image; I hope it's at least legible.

This proof that E(X + Y) = E(X) + E(Y) uses notation that I don't understand. P(X = x) refers to the probability of a single element of the random variable X, but I'm not sure what P(X = x, Y = y) means. Is it P(x and y)? P(x or y)? Something else? I'm also puzzled by the transition from (3) to (4), which indicates that the sigma and the P(Y = y) somehow cancel out.

The Probability Tutoring Book, Carol Ash, 1993, p. 78

Answer 1

$\{X=x, Y=y\}$ means that $X=x$ and $Y=y$.

From equation $(3)$ to equation $(4)$, the trick is

$$\sum_y P(X=x, Y=y)=P(X=x)$$

That is we consider all the possible values that $Y$ can take and sum it up.

This is due to $$\bigcup_y \{X=x, Y=y\}=\{X=x\}$$

and those sets on the left are disjoint.

Answer 2

$P(X=x,Y=y)$ is a shorthand for $P(\{X=x\}\cap\{Y=y\})$, that is probability that both events $\{X=x\}$ and $\{Y=y\}$ occur.

$\sum_y{P(X=x,Y=y)}=P(X=x)$ because the events $\{Y=y\}$ partition the sample space.