Counting Ordered Pairs

Proof

Every integer is uniquely represented in the form 2^pq, where p ≥ 0, q ≥ 1, q an odd number.

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

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

1. 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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