Equivalence Relation/Examples/Non-Equivalence/Greater Than

Example of Relation which is not Equivalence

Let $\R$ denote the set of real number.

Let $>$ denote the usual relation on $\R$ defined as:

$\forall \tuple {x, y} \in \R \times \R: x > y \iff \text {$x$ is (strictly) greater than $y$}$

Then $>$ is not an equivalence relation.


Proof

We have that $>$ is transitive:

$x > y, y > z \implies x > z$


But $>$ is not reflexive:

$\forall x: x \not > x$

$>$ is not symmetric:

$x > y \implies y \not > x$

So $\sim$ is not symmetric.


So $\sim$ is not an equivalence relation.

$\blacksquare$


Sources

  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.2$. Equivalence relations: Example $32$
  • 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations: Problem Set $\text{A}.2$: $7 \ \text{(b)}$
applied to a specific instance
  • 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.25$
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