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In [[mathematics]] a '''pairing function''' is a process to uniquely encode two [[natural number]]s into a single natural number.
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Any pairing function can be used in [[set theory]] to prove that [[integer]]s and [[rational number]]s 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'''<sup>''k''</sup> → '''N''' into a new function ''g'':'''N''' → '''N'''.
 
==Definition==
A '''pairing function''' is a [[primitive recursive function|primitive recursive]] [[bijection]]
:<math>\pi:\mathbb{N} \times \mathbb{N} \to \mathbb{N}.</math>
 
== Cantor pairing function ==
 
[[Image:Pairing natural.svg|thumb|The Cantor pairing function assigns one natural number to each pair of natural numbers]]
 
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 [[triangular number|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>.
where <math> \left\lfloor \, \right\rfloor</math> is the [[floor function]].
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 [[injective function|one-to-one]] and [[surjective function|onto]].
 
==References==
*{{mathworld|urlname=PairingFunction|author=Steven Pigeon|title=Pairing function}}
 
[[Category:Set theory]]

Latest revision as of 12:47, 8 April 2014

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Find your dream Singapore / JB Iskandar property or Selection Investment Properties for sale in London, Australia/Melbourne, Philippines and Cambodia at Singapore's leading property portal for new condominium launches and new commercial/industrial property launches Now we have the latest and latest condominiums / new condos for sale in Singapore , landed houses on the market in Singapore , as well as new retailers on the market, new workplaces for sale, and new restaurants on the market in Singapore. Find out Why Purchase New Condo Launches , or learn on for some of the scorching properties in the Featured Listings beneath.

Regent Quarter Residences - a collection of 34 one and two bed room residences and one penthouse on the market at the coronary heart of King's Cross. Just 4 minutes' walk from King's Cross St Pancras underground station, Regent Quarter lies at the heart of the exciting King's Cross regeneration area. We are hoping that you would give a call or an appointment after you have got gone by our website. Additionally it is a sincere wash from out side that you'd douche honour and provides us your actual property dealings. Once you hire us, you will discover that we are top 10 residential property developers in singapore (related web-site)-of-the-line actual estate businesses. You'll be able to you'll want to get glorious providers when employ us. BCA web site and data on resolving disputes on constructing defects after buy. freehold i.e. an absolute title;

Jun 18 Fashionable one bedroom house $one hundred ten / 1br - (SINGAPORE) pic actual property - by proprietor Jun 10 Modern one bedroom condominium $one hundred ten / 1br - (SINGAPORE) pic real estate - by proprietor Many Singapore London property buyers are hunting downproperties inLondon to take a position as SingDollar rose to document againststerling pound making it an inexpensive choice. Asidefrom being an excellent time for investment both for capitalappreciation and forex appreciation London properties are worthyinvestment as traders may look forwardto reaping good yields. Upcoming London property for Sale in Singapore Currencyexchange rate favours Singapore London Property Investors Property Status For Rent For Sale Open Home Extra search options Select - Condominium Rental Store home Rental Sort

8,000 houses near the American Faculty. Jalan Kasau bungalow for rent at $eight,000 with 3 American families in a small subdivision of 8 lots. A 5-min drive to the American School and 15-minute walk to the Kranji subway. Plenty of greenery, no site visitors and tranquility, however you do not get 500 children round this neighbourhood on Halloween's Day as you'll, in the subdivision of Woodgrove Property near the American College. Rentals have dropped considerably in 2003.

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