Set of Ordered Pairs of Integers is Countable Infinite
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Theorem
The set of all ordered pairs of integers $\Z$ is countably infinite.
Proof
The set of all ordered pairs of a set $S$ is by definition the Cartesian product $S \times S$.
In this context we are determining the cardinality of $\Z \times \Z$.
From Integers are Countably Infinite, we have that $\Z$ is a countably infinite set.
The result then follows from Cartesian Product of Countable Sets is Countable.
$\blacksquare$
Sources
- 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: Exercises: $1.11$