Bijection

A bijection is a function that is both

• one-to-one, and
• onto.

In other words, both injective and surjective.

The concept is frequently used to defined equivalence between mathematical structures:

• isomorphism,
• homeomorphism,
• diffeomeorphism, etc.

If there is a bijection $$f:X\rightarrow Y$$ between two set, we say that they have the same cardinality: $$|X|=|Y|.$$