Cardinality

Definition

The set $A$ has the same cardinality as $B$ if there exists $f:A \rightarrow B$ that is one-to-one and onto. In this case, we write $A \sim B$. (Abbott – Understanding_Analysis).

If there is an injection $g \rightarrow Y$, we say that the cardinality of $X$ is less than or equal to the cardinality of $Y$ and write $\# X \leq \# Y$. If $\# x \leq \# Y$ but $\# X \neq \# Y$, we say that $X$ is of strictly smaller cardinality than $Y$ and write $\# X < \# Y$ (in this this, no injection $g:X \rightarrow Y$ can be a surjection). (Schilling)