Equivalence Relation/Examples/Non-Equivalence/Greater Than
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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 $2.25$