How can I interpret $\max(X,Y)$? My textbook says:

Let $X$ and $Y$ be two stochastically independent, equally distributed
  random variables with distribution function F. Define $Z = \max (X, Y)$.

I don't understand what is meant by this. I hope I translated it correctly.
I would conclude that $X=Y$ out of this. And therefore $Z=X=Y$.
How can I interpret $\max(X,Y)$?

Think of it this way: to sample from $Z$, take one sample point from $X$ ($x_i$) and one from $Y$ ($y_i$). The point you sampled from $Z$ is $\max(x_i,y_i)$.
If by $X = Y$ you mean that $X$ and $Y$ are the same distribution AND they're perfectly correlated, then indeed $X = Y = Z$. Otherwise, no. Take for example $X$ and $Y$ to be independent, uniform distributions on $[0, 1]$. Then $Z = \max(X, Y)$ will follow a triangular distribution: $P(Z = z_i) = 2z_i$.
