Suppose that $A= \{a_1, \ldots, a_n\}$ and $B= \{b_1, \ldots, b_n\}$ are two sets of real numbers.
In such a situation, I usually find myself writing $\max\{a_i, b_i\mid i=1, \ldots, n\}$ as shorthand for the maximum of $\{a_1, \ldots, a_n, b_1, \ldots, b_n\}$.
I'm not sure I have seen this exact notation elsewhere. In general, is it an unambiguous way to express the maximum of the union of two finite sets of reals (with equal cardinality)?
Your notation is a little ambiguous, I suppose you intend to write $$\max \left(\bigcup_{i=1}^n \{a_i,b_i\}\right).$$
I would prefer to write $\max(A \cup B)$ instead.