Reflexive Relation/Examples/Reflexive Relation on Cartesian Plane

Examples of Use of Symmetric and Transitive Relation is not necessarily Reflexive
The subset of the Cartesian plane defined as:
 * $\RR := \set {\tuple {x, y} \in \R^2: x \le y \le x + 1}$

determines a relation on $\R^2$ which is reflexive, but neither symmetric nor transitive.

Reflexive Relation
We note that, by definition:
 * $\forall x \in \R: \tuple {x, y} \in \RR$ such that $x = y$

and so:


 * $\forall x \in \R: \tuple {x, x} \in \RR$

Hence $\RR$ is reflexive.

Non-Symmetric Relation
Proof by Counterexample:

But:
 * $1 > 0$

and so:
 * $\tuple {1, 0} \notin \RR$

thus demonstrating that $\RR$ is not symmetric.

Non-Transitive Relation
Proof by Counterexample:

but
 * $\tuple {0, 2} \notin \RR$

thus demonstrating that $\RR$ is not transitive.

The relation $\RR$ is illustrated below:


 * Reflexive-NonSymmetric-NonTransitive.png