Symmetric and Transitive therefore Reflexive

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Fallacy

Let $\RR \subseteq S \times S$ be a relation which is symmetric and transitive.

Then $\RR$ is also always reflexive.


Consider $x, y \in S$.

Suppose $x \mathrel \RR y$.

Then as $\RR$ is symmetric, it follows that $y \mathrel \RR x$.

As $\RR$ is transitive, it follows that $x \mathrel \RR x$.

Therefore $x \mathrel \RR x$ and so $\RR$ is reflexive.


Resolution

For $\RR$ to be reflexive, it is necessary for $x \mathrel \RR x$ for all $x \in S$.

Unless it is the case that $\forall x \in S: \exists y \in S: x \mathrel \RR y$, it is not necessarily the case that also $y \mathrel \RR x$, and so the reasoning does not follow.

Take the set $S = \set {0, 1}$ and the relation:

$\RR \subseteq S \times S: x \mathrel \RR y \iff x = y = 1$


It is seen that:

$\RR$ is symmetric
$\RR$ is transitive

but:

$\RR$ is not reflexive, as $\neg \paren {0 \mathrel \RR 0}$.

$\blacksquare$


Also see


Sources

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