Why use max for union and min for intersection?

Answer 1

Because they obey the same laws (specifically, de Morgan's Laws): -max(a,b) = min(-a,-b) and -min(a,b) = max(-a,-b). In theory, you could use max for intersection and min for union, too.

Answer 2

The reason we use max for union is because the union of two sets should include all elements from both sets, so the max operation ensures that the larger of the two elements is included. Conversely, we use min for intersection because we want to find the common elements shared by both sets, so the min operation ensures we only include elements that are in both sets.