A **bijection** is a type of function in mathematics that establishes a one-to-one correspondence between elements of two sets. A function \( f: A \to B \) is called a bijection if it satisfies two main properties: 1. **Injective (One-to-One):** For every pair of distinct elements \( a_1, a_2 \in A \), \( f(a_1) \neq f(a_2) \).
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A bijection is a function that is both injective and surjective: each element of its codomain is the output of exactly one element of its domain of a function. The two requirements exclude unused targets and merged inputs respectively. The finite example , , is a bijection from to , whereas adding an unused target makes that same rule fail surjectivity.
A bijection gives an inverse function by reversing every pairing. Conversely, a two-sided inverse function implies a bijection, as the proof under inverse function shows. This is useful for transporting structures without losing or identifying elements. It also defines equality of cardinality: two sets have equal cardinality exactly when a bijection pairs them. An infinite example is from the integers to the even integers, with inverse on that specified codomain. A proper subset can therefore have the same cardinality as an infinite set. [ref]
https://discrete.openmathbooks.org/preview/sec_intro-functions.html. Levin, functions: injections, surjections and bijections
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