Suppose $X,Y$ are random variables. I'm trying to understand why $\max\{X,Y\}$ and $\min\{X,Y\}$ are also random variables. The proof in the book that I'm using states that for each $t$,
$\{ \max\{X,Y\} \leq t \} = \{X \leq t \} \cap \{Y \leq t \}$,
and
$\{ \min\{X,Y\} \leq t \} = \{X \leq t \} \cup \{Y \leq t \}$.
I can't make the mental leap. Could someone explain this more explicitly?
Edit:
We use the following definition for a random variable:
A random variable is a function on $X: \Omega \rightarrow \mathbb{R}$ such that
$X^{-1}((-\infty,t]) := \{\omega \in \Omega : X(\omega) \leq t\} \in \mathcal{F}$
for all $t\in\mathbb{R}$, where $\mathcal{F}$ is the set of all events.
The obvious first step that I made was
$\{ \max\{X,Y\} \leq t \} = \{ \omega : \max\{X(\omega),Y(\omega)\} \leq t \}$.
I just don't see why the next equality follows.
Edit2: Fixed typo.
Best Answer
It's rather $Y\le t$ instead of $Y\le y$. With that, for an $\omega$ we have $$\omega\in\{\max(X,Y)\le t\} \iff (X(\omega)\le t)\land(Y(\omega)\le t) \\ \iff \omega\in\{X\le t\}\cap\{Y\le t\}\,.$$ Similarly for the $\min$.