Maximum $m:=\max${$u(x)$ s.t $x \in A\bigcup B$}

Question

How can we express $m$ denoted by $m:=\max${$u(x)$ s.t $x \in A\bigcup B$} as comparing it with $\max${$u(x)$ s.t $x \in A$} and $\max${$u(x)$ s.t $x \in B$}

What about minimum? I need only clarification, you can without proof if you have no time.

Answer 1

Denote by $m_A:= \max\{u(x)\colon x\in A\}$ and by $m_B:= \max\{u(x)\colon x\in B\}$. Then what you are looking for is the equation $m = \max\{m_A, m_B\}$.

The equality of right and left hand side can be shown by showing both $\leq$ and $\geq$.

The case of the minimum works analoguously, but for minima instead of maxima.