Cross-Relation on Real Numbers is Equivalence Relation/Geometrical Interpretation

Theorem

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

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

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

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

Proof

We have from Cross-Relation on Real Numbers is Equivalence Relation that $\alpha$ is an equivalence relation.

Thus:

\(\ds \tuple {x_1, y_1}\)

\(\alpha\)

\(\ds \tuple {x_2, y_2}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x_1 + y_2\)

\(=\)

\(\ds x_2 + y_1\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x_1 - y_1\)

\(=\)

\(\ds x_2 - y_2\)

Thus each equivalence classes consists of sets of points such that:

$x - y = c$

That is:

$y = x + c$

Thus from Equation of Straight Line in Plane, this is the equation of a straight line whose slope is $1$.

$\blacksquare$

Sources

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