Pairing function

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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:NkN into a new function g:NN.

Definition[edit]

A pairing function is a primitive recursive bijection

Cantor pairing function[edit]

The Cantor pairing function assigns one natural number to each pair of natural numbers

The Cantor pairing function is a pairing function

defined by

The statement that this is the only quadratic pairing function is known as the Fueter–Pólya theorem. Whether this is the only polynomial pairing function is still an open question. 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

as

Inverting the Cantor pairing function[edit]

Let be arbitrary and suppose that . We will show that there exist unique values such that

and hence that is invertible. 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

and thus

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.

Examples[edit]

To calculate π(47, 32):

  • 47 + 32 = 79
  • 79 + 1 = 80
  • 79 × 80 = 6320
  • 6320 ÷ 2 = 3160
  • 3160 + 32 = 3192

so π(47, 32) = 3192.

To find x and y such that π(x, y) = 1432:

  • 8 × 1432 = 11456
  • 11456 + 1 = 11457
  • √11457 = 107.037
  • 107.037 − 1 = 106.037
  • 106.037 ÷ 2 = 53.019
  • ⌊53.019⌋ = 53

so w = 53

  • 53 + 1 = 54
  • 53×54 = 2862
  • 2862÷2 = 1431

so t = 1431

  • 1432 − 1431 = 1

so y = 1

  • 53 − 1 = 52

so x = 52; thus π(52, 1) = 1432.

References[edit]

External links[edit]