Explanation of $\{ S\wedge{T}\leq{t}\} =\{S\leq t\} \cup \{T\leq t\}$

Question

From: Cohen, Elliott. "Stochastic Calculus and Applications".

$\{ S\wedge{T}\leq{t}\} =\{S\leq t\} \cup \{T\leq t\}$ For fix t, take min{S,T}, for example S, and set $\{\omega:S(\omega)\}$. Why it equals union not intersection? Could you give an example? My imagination suggest that it should be intersection. For the second assertion $infT_n$, why do we consider strict case in the beginning?

Answer 1

$$\{S \land T \le t\} = \{\min\{S,T\} \le t\} = \{\text{at least one of $S$ and $T$ is $\le t$}\} = \{S \le t\} \cup \{T \le t\}.$$

$$\{S \lor T \le t\} = \{\max\{S,T\} \le t\} = \{\text{both $S$ and $T$ are $\le t$}\} = \{S \le t\} \cap \{T \le t\}.$$