Definition. Symmetry [dm-000Y]

Liam O'Connor

A binary relation $\mathcal {R}$ is symmetric iff, for all $x$ and $y$ , $x\ \mathcal {R}\ y$ implies $y\ \mathcal {R}\ x$ .

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Equivalence relation [dm-000W]

Liam O'Connor

A binary relation $(\approx ) \subseteq X \times X$ is an equivalence relation if it is, for all $x, y, z \in X$ :

reflexive: $x \approx x$ ,
transitive: $x \approx y$ and $y \approx z$ implies $x \approx z$ ,
symmetric: $x \approx y$ implies $y \approx x$