# Symmetric and Transitive therefore Reflexive

## Contents

## Fallacy

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

Then $\mathcal R$ is also always reflexive.

Consider $x, y \in S$.

Suppose $x \mathrel {\mathcal R} y$.

Then as $\mathcal R$ is symmetric, it follows that $y \mathrel {\mathcal R} x$.

As $\mathcal R$ is transitive, it follows that $x \mathrel {\mathcal R} x$.

Therefore $x \mathrel {\mathcal R} x$ and so $\mathcal R$ is reflexive.

## Resolution

For $\mathcal R$ to be reflexive, it is necessary for $x \mathrel {\mathcal R} x$ for **all** $x \in S$.

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

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

- $\mathcal R \subseteq S \times S: x \mathrel {\mathcal R} y \iff x = y = 1$

It is seen that:

- $\mathcal R$ is symmetric
- $\mathcal R$ is transitive

but:

- $\mathcal R$ is not reflexive, as $\neg \paren {0 \mathrel {\mathcal R} 0}$.

$\blacksquare$

## Also see

- Definition:Equivalence Relation, which is the usual motivator of this frequently-met fallacy.

- Symmetric Transitive and Serial Relation is Reflexive which shows that the condition under which a symmetric and transitive relation
*is*guaranteed to be reflexive.

## Sources

- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): Exercise $1.2: 6$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.3$: Example $5$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{I}$: Exercise $\text{F}$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Exercise $6$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.4$: Equivalence relations: Exercise $4$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations: Exercise $3.3$