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 $1$.

$\blacksquare$


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