# Example of a relation that is reflexive, symmetric, antisymmetric but not transitive.

Please, can you help a beginner mathematician with the following problem?

Is there a binary relation that is reflexive, symmetric, antisymmetric but not transitive?

# Relation

Let be two sets $$A$$,$$B$$ $$\neq\emptyset$$. A relation $$\mathscr{R}$$ of $$A$$ to $$B$$ is the ordered triple ($$A$$,$$B$$,$$\mathscr{R}$$) where $$\mathscr{R}$$ $$\subset$$ $$A\timesB$$, $$A$$ is called input set, $$B$$ is called output set and $$\mathscr{R}$$ is called matching rule or graphic.

Note: A particular case of relation is when the input set and output set are equal i.e. $$A$$=$$B$$.
Let $$A$$ $$\neq\emptyset$$. Hereinafter, we say that $$\mathscr{R}$$ it is a relation of $$A$$ to $$A$$. Furthermore, $$(a,b)\in$$ $$\mathscr{R}$$, then we will denote $$a$$ $$\mathscr{R}$$ $$b$$.

### Reflexive

A relation $$\mathscr{R}$$ is called reflexive iff: $$\forall x\in A:$$ $$x$$ $$\mathscr{R}$$ $$x$$.

### Symmetric

A relation $$\mathscr{R}$$ is called symmetric iff: $$\forall x,y\in A:$$ $$x$$ $$\mathscr{R}$$ $$y$$ $$\Rightarrow$$ $$y$$ $$\mathscr{R}$$ $$x$$.

### Transitive

A relation $$\mathscr{R}$$ is called transitive iff: $$\forall x,y,z\in A:$$ $$x$$ $$\mathscr{R}$$ $$y$$ $$\wedge$$ $$y$$ $$\mathscr{R}$$ $$z$$ $$\Rightarrow$$ $$x$$ $$\mathscr{R}$$ $$z$$.

### Antisymmetric

A relation $$\mathscr{R}$$ is called antisymmetric iff: $$\forall x,y\in A:$$ $$x$$ $$\mathscr{R}$$ $$y$$ $$\wedge$$ $$y$$ $$\mathscr{R}$$ $$x$$ $$\Rightarrow$$ $$x$$ $$=$$ $$y$$

If the answer is true, then please show me a couple of examples.
Thank you.

Quote of the day:

"There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world".

Nicolai Ivanovitch Lobachevsky
1792-1856

• As I told you, it can be only isolated points.
– L F
Nov 1, 2016 at 22:58
• Oh, I understand. Please, could you explain with a couple of examples? Nov 1, 2016 at 23:06
• The answer of Alephnull shows that a relation with your characteristics is nessesary identity, which is transitive. So you can't ask for non transivity
– L F
Nov 1, 2016 at 23:13
• Pretty sure Nicolai Lobachevsky is wrong... Nov 2, 2016 at 2:02
• Oct 25, 2021 at 13:14