Definition
To have an exact pairing between X and Y (where Y need not be different from X), four properties must hold:
- each element of X must be paired with at least one element of Y,
- no element of X may be paired with more than one element of Y,
- each element of Y must be paired with at least one element of X, and
- no element of Y may be paired with more than one element of X.
Satisfying properties (1) and (2) means that a bijection is a function with domain X. It is more common to see properties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y. Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions). Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injective functions). With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both one-to-one and onto.
Read more about this topic: Bijection
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