Bijection

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math

A function that is both bijective and surjective, I.E. a one-to-one [see page 10, mapping] function. That is to say a function which can map from its domain to the range and from the range back to the domain. Every input maps to a unique output and every output has a unique input.

Note: Every bijection has an inverse, making them invaluable for encoding data.

For a [see page 7, system] with $n$-elements in the domain and $n$-elements in the range we can form up-to \( n! \) possible bijections (possible permutations of the mappings) because:

• The first element of the domain can map to one of the \( n \) elements of the range.
• Leaving \( n-1 \) elements of domain to map to one of the \( n-1 \) elements in the range.
• Leading to \( n \times (n-1) \times (n-2) \times (n-3) \times \ldots = n! \forall n \ge 0 \).