Please, can you help a beginner mathematician with the following problem?
Is there a binary relation that is reflexive, symmetric, antisymmetric but not transitive?
Definitions:
Relation
Let be two sets $A$,$B$ $\neq$$\emptyset$. A relation $\mathscr{R}$ of $A$ to $B$ is the ordered triple ($A$,$B$,$\mathscr{R}$) where $\mathscr{R}$ $\subset$ $A$$\times$$B$, $A$ is called input set, $B$ is called output set and $\mathscr{R}$ is called matching rule or graphic.
Note: A particular case of relation is when the input set and output set are equal i.e. $A$=$B$.
Let $A$ $\neq$$\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
Best Answer
Assume we have such a relation. It is symmetric so xRy implies yRx. It is antisymmetric so xRy and yRx implies x=y. But putting this together we get xRy implies x=y. Thus our relation is the identity function over some set. But the identity function is transitive vacuously. This is a contradiction.