Symmetric and Transitive Relation is not necessarily Reflexive

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Theorem

Let $S$ be a set.

Let $\alpha \subseteq S \times S$ be a relation on $S$.

Let $\alpha$ be both symmetric and transitive.


Then it is not necessarily the case that $\alpha$ is also reflexive.


Proof 1

Proof by Counterexample:

Let $S = \set {a, b, c}$.

Let:

$\alpha = \set {\tuple {a, a}, \tuple {a, b}, \tuple {b, a}, \tuple {b, b} }$

By inspection it is seen that $\alpha$ is both symmetric and transitive.

However, we have:

$\neg c \mathrel \alpha c$

Hence $\alpha$ is both symmetric and transitive but not reflexive.

$\blacksquare$


Proof 2

Proof by Counterexample:

Let $S = \Z$ be the set of integers.


Let $\alpha$ be the relation on $S$ defined as:

$\forall x, y \in S: x \mathrel \alpha y \iff x = y = 0$


\(\ds x\) \(\alpha\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds y = 0\)
\(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds x = 0\)
\(\ds \leadsto \ \ \) \(\ds y\) \(\alpha\) \(\ds x\)

Thus $\alpha$ is symmetric.


Now let $x \mathrel \alpha y$ and $y \mathrel \alpha z$

Then:

$x = y = 0, y = z = 0$

and so

$x \mathrel \alpha z$


Now let $x = \Z$ such that $x \ne 0$.

Then it is not the case that:

$x \mathrel \alpha x$

and so $\alpha$ is not reflexive.


Hence $\alpha$ is both symmetric and transitive but not reflexive.

$\blacksquare$


Proof 3

Proof by Counterexample:

Let $S = \set {1, 2}$ be a set.


Let $\RR$ be the relation on $S$ defined as:

$\forall x, y \in S: x \mathrel \RR y \iff x = y = 2$


\(\ds x\) \(\RR\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds y = 2\)
\(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds x = 2\)
\(\ds \leadsto \ \ \) \(\ds y\) \(\RR\) \(\ds x\)

Thus $\alpha$ is symmetric.


Now let $x \mathrel \RR y$ and $y \mathrel \RR z$

Then:

$x = y = 2, y = z = 2$

and so:

$x \mathrel \RR z$


Now let $x = 1$.

Then it is not the case that:

$x \mathrel \RR x$

and so $\RR$ is not reflexive.


Hence $\RR$ is both symmetric and transitive but not reflexive.

$\blacksquare$


Examples

Subset of Cartesian Plane

The subset of the Cartesian plane defined as:

$\RR := \set {\tuple {x, y} \in \R^2: -1 \le x \le 1, -1 \le y \le 1}$

determines a relation on $\R^2$ which is symmetric and transitive but not reflexive.


Sources

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