Cross-Relation is Equivalence Relation

Theorem
Then $\boxtimes$ is an equivalence relation on $\struct {S_1 \times S_2, \oplus}$.

Reflexivity

 * $\forall \tuple {x_1, y_1} \in S_1 \times S_2: x_1 \circ y_1 = x_1 \circ y_1 \implies \tuple {x_1, y_1} \boxtimes \tuple {x_1, y_1}$

So $\boxtimes$ is a reflexive relation.

Symmetry
So $\boxtimes$ is a symmetric relation.

Transitivity
So $\boxtimes$ is a transitive relation.

All the criteria are therefore seen to hold for $\boxtimes$ to be an equivalence relation.

Examples

 * Cross-Relation on Natural Numbers is Equivalence Relation
 * Cross-Relation on Real Numbers is Equivalence Relation