Cross-Relation on Natural Numbers is Equivalence Relation
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Theorem
Let $\struct {\N, +}$ be the semigroup of natural numbers under addition.
Let $\struct {\N \times \N, \oplus}$ be the (external) direct product of $\struct {\N, +}$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$.
The relation $\boxtimes$ defined on $\N \times \N$ by:
- $\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$
is an equivalence relation on $\struct {\N \times \N, \oplus}$.
Proof
$\boxtimes$ is an instance of a cross-relation.
We also have that Natural Number Addition is Commutative.
The result therefore follows from Cross-Relation is Equivalence Relation.
$\blacksquare$
Sources
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 2.5$: Theorem $2.20$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Mappings: Exercise $9$