I have a confusion about stopping times and hope you can help me there!
Let $\sigma$ and $\tau$ be stopping times with values in $I=\big[0,\infty)$ with respect to the filtration $\mathbb{F} = (\mathcal{F}_t)_{t \in I}$.
So for $t \in I$ by definition we have $\{\tau \leq t \} = \{\omega \in \Omega: \tau(\omega) \leq t\} = A_{\tau} \in \mathcal{F}_t$ and $\{\sigma \leq t \} = \{\omega\in \Omega: \sigma(\omega) \leq t\} = A_{\sigma} \in \mathcal{F}_t$. My book now says that the minimum $\{\tau \wedge \sigma \leq t \} = \{\tau \leq t \}\cup \{\sigma \leq t \}$ and the maximum $\{\tau \vee \sigma \leq t \} = \{\tau \leq t \}\cap \{\sigma \leq t \}$. Why are they union and intersection of the stopping times?