In mathematics a pairing function is a process to uniquely encode two natural numbers into a single natural number.
Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. In theoretical computer science they are used to encode a function defined on a vector of natural numbers f:Nk → N into a new function g:N → N.
A pairing function is a primitive recursive bijection
Cantor pairing function
The Cantor pairing function assigns one natural number to each pair of natural numbers
The Cantor pairing function is a pairing function
When we apply the pairing function to and we often denote the resulting number as
This definition can be inductively generalized to the Cantor tuple function
Inverting the Cantor pairing function
Suppose we are given z with
and we want to find x and y. It is helpful to define some intermediate values in the calculation:
where t is the triangle number of w. If we solve the quadratic equation
for w as a function of t, we get
which is a strictly increasing and continuous function when t is non-negative real. Since
we get that
where is the floor function.
So to calculate x and y from z, we do:
Since the Cantor pairing function is invertible, it must be one-to-one and onto.