I am puzzled. Wikipedia says: "|X| ≤ |Y| means that there exists an injective function from X to Y." Let's see sets A and B: A = {1,2,3} and B = {1,2}. f: A → B: 1 ↦ 1, 2 ↦ 2. f is injective, but |B| ≤ |A|. Did I understand something wrong?
2026-03-28 12:02:53.1774699373
Order of cardinal number
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The function you describe is a valid function, and an injection, from $B$ to $A$ (all members of $B$ have a value in $A$), but not a valid function on $A$, as $3$ has no value in $B$. And we cannot give it a value without killing the injectivity.... One easily checks that no function from $A$ to $B$ can be an injection. So $|B| \le |A|$ but not reversely.