Consider the following two statements (where $[X]^2$ denotes the set of all unordered two-element subsets of $X$):
(HSO) For every infinite set $X$ there exists an injection $f: X \times X \hookrightarrow X$
(HSU) For every infinite set $X$ there exists an injection $f: [X]^2 \hookrightarrow X$
The following is an exercise from a book I'm currently reading:
So I thought I could argue as follows: We know that there are more ordered pairs ($|X|^2$) than unordered pairs (${|X| \choose 2}$), hence there exists an injection $i: [X]^2 \hookrightarrow X \times X$ hence $f \circ i$ is an injection $[X]^2 \hookrightarrow X$. But since the exercise is rated uber-difficult, this argument must be flawed. Can you tell me what's wrong with it? Thanks a lot.