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 $n\times (n-1)\times (n-2)\times (n-3)\times \dots =n!\mathrm{\forall }n\ge 0$.