Cross-Relation on Real Numbers is Equivalence Relation

Theorem

Let $\R^2$ denote the cartesian plane.

Let $\alpha$ denote the relation defined on $\R^2$ by:

$\tuple {x_1, y_1} \mathrel \alpha \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$

Then $\alpha$ is an equivalence relation on $\R^2$.

Geometrical Interpretation

The equivalence classes of $\alpha$, when interpreted as points in the plane, are the straight lines of slope $1$.

Proof

$\alpha$ is an instance of a cross-relation.

We also have that Real Addition is Commutative.

The result therefore follows from Cross-Relation is Equivalence Relation.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $4$