In set theory, $A \cup B$ is logically defined as $\{x : x \in A \lor x \in B\}$. In set theory, the result of unionizing A with B is a bigger set, but in logic, "or" is a softening operation.
In set theory, $A \cap B$ is logically defined as $\{x : x \in A \land x \in B\}$. In set theory, the result of intersecting A with B is a smaller set, but in logic, "and" is a strengthening operation.
Why do the extensional results of union and intersection go in reverse from their logical definition?