Symmetric and Transitive Relation is not necessarily Reflexive/Examples/Subset of Cartesian Plane

Examples of Use of Symmetric and Transitive Relation is not necessarily Reflexive
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.

Non-Reflexive Relation
We note that, for example, $\tuple {2, 2} \notin \RR$.

Hence $\RR$ is non-reflexive.

Symmetric Relation
thus demonstrating that $\RR$ is symmetric.

Transitive Relation
thus demonstrating that $\RR$ is transitive.

The relation $\RR$ is illustrated below:


 * Symmetric-Transitive-NonReflexive.png