# Counting Ordered Pairs

The set of ordered pairs of elements of a countable set is countable.

### Proof

Every integer is uniquely represented in the form 2^{p}q, where

For a pair (m, n) ∈ **N**×**N**, where **N** is the set of natural numbers, define

f(m, n) = 2

^{m - 1}(2n - 1).Function f is a bijection from **N**×**N** to **N**. (It is obviously 1-1. It is onto because of the sentence that opens the proof.)

That's it.

### References

- DAVID M. BRADLEY,
__Counting Ordered Pairs__,*Math magazine*,**83**(2010) 302

### Countability of Rational Numbers

- Counting Ordered Pairs
- Countable Times Countable Is Countable
- Countability of Rational Numbers
- Countability of Rational Numbers: PWW
- Countability Principle
- Countability of Rational Numbers via Conitnued Fractions
- Fractions on a Binary Tree II
- Countability of Rational Numbers as a Union of Finite Sets
- A New Proof That the Rationals Are Countable

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