Definition:Bijection

Definition
A mapping $$f: S \to T$$ is a bijection or bijective or a one(-to)-one correspondence iff $$f$$ is both a surjection and an injection.

If a bijection exists between two sets $$S$$ and $$T$$, then $$S$$ and $$T$$ are said to be in one-to-one correspondence.

It is clear that a bijection is a relation which is:
 * left-total;
 * right-total;
 * functional;
 * injective.

Basic Properties of a Bijection

 * In Bijection iff Left and Right Inverse, it is shown that a mapping $$f$$ is a bijection iff it has both a left inverse and a right inverse, and that these are the same, called the two-sided inverse.


 * In Bijection iff Inverse is Bijection, it is shown that the inverse mapping $$f^{-1}$$ of a bijection $$f$$ is also a bijection, and that it is the same mapping as the two-sided inverse.


 * In Bijection Composite with Inverse, it is established that the inverse mapping $$f^{-1}$$ and the two-sided inverse are the same thing.


 * In Bijection iff Left and Right Cancellable, it is shown that a mapping $$f$$ is a bijection iff it is both left cancellable and a right cancellable.

Also see

 * Injection
 * Surjection