Size Comparisons for Sets

Definition: Cardinality Comparisons

Let $A$ and $B$ be two sets.

We say that:

• the cardinality of $A$ is equal to the cardinality of $B$, written as $|A|=|B|$, if there is a bijection from $A$ to $B$;
• the cardinality of $A$ is less than or equal to the cardinality of $B$, written as $|A|\le |B|$, if there is an injection between $A$ and $B$;
• the cardinality of $A$ is less than the cardinality of $B$, written as $|A|<|B|$, if there is an injection but no bijection from $A$ to $B$.