Question

Show that if A and B are sets with the same cardinality, then |A|<=|B| and |B|<=|A|

Answer 1

We say that $|X|\leq |Y|$ if there is an injection $i: X\to Y$. If the sets $A$ and $B$ are of the same cardinality, then there exists a bijection $f:A\to B$. Since each bijection is an injection, $f$ is an injection, and therefore $|A|\leq |B|$. The inverse function $f^{-1}:B\to A$ is a bijection as well. Consequently, $f^{-1}$ is an injection. We conclude that $|B|\leq |A|$, and we are done.