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 $\max(X,Y)$?

A random variable is a function from a sample space to $\mathbb{R}$. Therefore $Z=\max(X,Y)$ means $Z(p):=\max(X(p), Y(p))$ for each $p$ in the sample space.
Also, from that you can't conclude $X=Y=Z$.
