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

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## 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:

\(\displaystyle \tuple {x_1, y_1}\) | \(\alpha\) | \(\displaystyle \tuple {x_2, y_2}\) | |||||||||||

\(\displaystyle \iff \ \ \) | \(\displaystyle x_1 + y_2\) | \(=\) | \(\displaystyle x_2 + y_1\) | ||||||||||

\(\displaystyle \iff \ \ \) | \(\displaystyle x_1 - y_1\) | \(=\) | \(\displaystyle x_2 - y_2\) |

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 $4$