A and B have the same cardinality if there is a 1-1 mapping of A onto B (bijective).

If there is a 1-1 mapping of A onto a subset of B (the same cardinality), there exists a 1-1 mapping into B (injective).

If there is an injective mapping into a subset of B, The mapping is onto that subset from A, and A and that subset have the same cardinaity.