Final answer:
The size of the image of a subset S under a function f is always less than or equal to the size of S itself, because a function can map multiple elements of S to the same element in B, but not vice versa.
Step-by-step explanation:
The question revolves around why the size of the image of a subset S under a function f, denoted as |f(S)|, is always less than or equal to the size of the subset S itself, when S is a subset of a finite set A, and f is a mapping from A to a finite set B. This statement holds true because a function can only map each element of S to a single element in B, but multiple elements in S can be mapped to the same element in B. Therefore, the number of distinct elements in f(S) can never exceed the number of elements in S, and in mathematical terms, this is expressed as |f(S)| ≤ |S|.