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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:N^k → N into a new function g:N → N.

Every pairing function is primitive recursive.

Contents 1 Definition 2 Cantor pairing function 2.1 Inverting the Cantor pairing function 3 References

Definition

A pairing function is a bijective function

<math>\pi:\mathbb{N} \times \mathbb{N} \to \mathbb{N}.</math>

Cantor pairing function

The Cantor pairing function is a pairing function

<math>\pi:\mathbb{N} \times \mathbb{N} \to \mathbb{N}</math>

defined by

<math>\pi(k_1,k_2) := \frac{1}{2}(k_1 + k_2)(k_1 + k_2 + 1)+k_2.</math>

When we apply the pairing function to <math>k_1</math> and <math>k_2</math> we often denote the resulting number as <math>\langle k_1, k_2 \rangle</math>

This definition can be inductively generalized to the Cantor tuple function

<math>\pi^{(n)}:\mathbb{N}^n \to \mathbb{N}</math>

as

<math>\pi^{(n)}(k_1, \ldots, k_{n-1}, k_n) := \pi ( \pi^{(n-1)}(k_1, \ldots, k_{n-1}) , k_n)</math>

Inverting the Cantor pairing function

Suppose we are given z with

<math> z = \langle x, y \rangle = \frac{(x + y)(x + y + 1)}{2} + y </math>

and we want to find x and y. It is helpful to define some intermediate values in the calculation:

<math> w = x + y \!</math>

<math> t = \frac{w(w + 1)}{2} = \frac{w^2 + w}{2} </math>

<math> z = t + y \!</math>

where t is the triangle number of w. If we solve the quadratic equation

<math> w^2 + w - 2t = 0 \!</math>

for w as a function of t, we get

<math> w = \frac{\sqrt{8t + 1} - 1}{2} </math>

which is a strictly increasing and continuous function when t is non-negative real. Since

<math> t \leq z = t + y < t + (w + 1) = \frac{(w + 1)^2 + (w + 1)}{2} </math>

we get that

<math> w \leq \frac{\sqrt{8z + 1} - 1}{2} < w + 1 </math>

and thus

<math> w = \left\lfloor \frac{\sqrt{8z + 1} - 1}{2} \right\rfloor </math>.

So to calculate x and y from z, we do:

<math> w = \left\lfloor \frac{\sqrt{8z + 1} - 1}{2} \right\rfloor </math>

<math> t = \frac{w^2 + w}{2} </math>

<math> y = z - t \!</math>

<math> x = w - y \!</math>.

Since the Cantor pairing function is invertible, it must be one-to-one and onto.

References

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• New Jersey Department of the Public Advocate | FAQs (see How can I use conservation to reduce utility bills? Energy conservation can help a great deal in reducing your utility bill.
• How can I Reduce my Utility Bills? Brief and Straightforward Guide: How can I Reduce my Utility Bills?
• Reduce Utility Costs We will analyze your average utility bill to help you reduce utility costs. Q: How do I know if I’m paying too much on my utility bills?
• Metrix Software You can track utility bills for your facilities, monitor performance, reduce utility cost overruns, and prioritize your facility management resources.
• myGHBA.com: Reduce Utility Bills While Increasing Comfort Concerned homeowners are looking for cost-effective ways to reduce utility bills while increasing the comfort of their homes.

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