# Equivalence Relation: Property 1 is implied by Properties 2 and 3 [duplicate]

Let $$x \sim y$$ be defined as meaning that the ordered tuple $$(x, y)$$ is in some set $$S$$. If this is the case, an equivalence relation on the set $$S$$ is defined as a subset of $$S \times S$$ with the following properties:

1. For all $$x$$ in $$S$$, $$x \sim x$$.
2. If $$x \sim y$$, then $$y \sim x$$.
3. If $$x \sim y$$ and $$y \sim z$$, then $$x \sim z$$.

I think this is an over-determined definition, because (1) is a consequence of (2) and (3), as long as it contains anything in addition to the empty set:

Assume that the equivalence relation contains at least point, $$x \sim y$$. By (2), it contains $$y \sim x$$. By (3), since $$x \sim y$$ and $$y \sim x$$, we also have $$x \sim x$$. Since we put no constraints on $$x$$ except that it be a part of the relation, this implies that for any point $$x \sim y$$, we have $$x \sim x$$. (We also have $$y \sim y$$, since $$x \sim y$$ implies $$y \sim x$$.) So why do we need to specify (1)?

• What you have shown is that if $\sim$ satisfies (2) and (3), then for every $x$ such that $x\sim y$ holds for some $y$ we have $x\sim x$. But the statement (1)': "For all $x$ there is some $y$ such that $x\sim y$" is a nontrivial hypothesis on $\sim$. FWIW, on its own (1)' is implied by but not implied by (1), but under the assumptions (2) and (3) they are equivalent. – Noah Schweber Sep 27 '19 at 17:49
• Any subset of the diagonal can be recognized as a symmetric and transitive relation. – drhab Sep 27 '19 at 18:35

Your error comes when you assume the relation is nonempty. The empty relation satisfies $$(2)$$ and $$(3)$$ vacuously, but not $$(1)$$. In fact the relation could be nonempty and still fail. Let the $$S=\{1,2,3\}$$ and the relation be $$\{(1,1),(2,2)\}$$. Again the relation satisfies $$(2)$$ and $$(3)$$, but not $$(1)$$
• The empty set is empty. It contains none of the pairs of $S \times S$, so no element is related to any other. Symmetry is stated as $x \sim y \implies y \sim x$, so if the relation is empty, the antecedent is always false and the implication is always true. – Ross Millikan Sep 27 '19 at 18:49