Let S and T be a (finite or infinite) sets. |S| and |T| denote their
We say that |S|?|T| (the cardinality of S is less or equal to the
cardinality of T) iff there is an injection from S to T.
If |S|?|T| but not |S|=|T|, we say that |S|<|T| (the cardinality of S
is strictly smaller than the cardinality of T).
We say that |S| = |T| (the cardinalities of S and T are equal) iff
there is a bijection from S to T.
Fact: |S| = |T| iff |S|?|T| and |T|?|S|.
The “only if” part is evident; the “if” part requires a special proof.