Symmetric Relation/Examples/Distance Less than 1

Example of Symmetric Relation

Let $\sim$ be the relation on the set of real numbers $\R$ defined as:

$x \sim y \iff \size {x - y} < 1$

Then $\sim$ is symmetric and reflexive, but not transitive.

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 7$: Relations