Distribution of $F_{Z}(z)$ where $Z = min\{X,Y\}$

Question

I'm having trouble understanding a step in my teacher's explanation.

$F_{Z}(z) =\mathbb{P}(Z\leq z) = \mathbb{P}(min\{X,Y\} \leq z) = 1 - \mathbb{P}(max\{X,Y\} > z) = 1 - \mathbb{P}(X>z, Y>z)$

I don't get why the complement of $\{min\{X,Y\}\leq z\}$ is $\{max\{X,Y\} > z\}$. Shouldn't it be $\{min\{X,Y\} > z\}$?

I suppose not, because by assuming that I ended up obtaining the distribution of $Z = max\{X,Y\}$.

But, for example, if $x < z$ and $y > z$, then $max\{x,y\} = y > z$ and $min\{x,y\} = x < z$. So how could one be the complement of the other, if both can be true at the same time?

Thanks.

Answer 1

The complement is $\{\min \{X,Y\} >z\}$ which is same as $\{X >z\, and \,Y>z\}$