Find a relation which is reflexive and symmetric but not transitive on integers

Question

The question is stated as: Let $A$ be the set of integers, find a relation $R$ which is reflexive and symmetric in $A$ but not transitive in $A$.

By definition we have that.

• $R$ is reflexive in $A$$ \Leftrightarrow (\forall x)(x \in A \Rightarrow xRx)$
• $R$ is symmetric in $A$$ \Leftrightarrow (\forall x)(\forall y)([x\in A \land y \in A \land xRy] \Rightarrow yRx)$
• $R$ is transitive in $A$$ \Leftrightarrow (\forall x)(\forall y)(\forall z)([x\in A \land y \in A \land z \in A \land xRz \land zRy] \Rightarrow xRy)$

What i thinked is to define such a relation using least common multiple, and greatest of two numbers as the following:

• Let $lcm(x,y)$ be the least common multiple of $x$ and $y$
• Let $max(x,y)$ be the greatest number from $\{x,y\}$
• Then let $R = \{(x,y) : x \in A \land y \in A \land lcm(x,y) = max(x,y) \}$

It is transitive because $(\forall x)(x \in A \Rightarrow lcm(x,x) = x = max(x,x))$.

It is symmetric too because if the if $lcm(x,y) = max(x,y)$ holds true, its obvious that $lcm(y,x) = max(y,x)$ will be true too for any integers.

But it is not transitive, i tried to show this with one counter example: $(6,3) \in R \land (3,9) \in R$ but $(6,9) \notin R$.

The way I defined the relation is correct? Its possible to retrieve relations from numerical sets holding choosed properties in a easy way?

Answer 1

One simple example:

$$a R b\iff ab\not\equiv 3\mod 10$$

$aRa$ because $a^2$ cannot end with digit 3.

$aRb \iff bRa$, because $ab=ba$.

But this relation is not transitive. For example, for $a=3$, $b=5$, $c=11$ we have $aRb$, $bRc$ but not $aRc$.

EDIT: Actually the simplest example I have found so far is this one:

$$a R b\iff a+b \ne101 $$

Obviously, it's reflexive, symmetric but not transitive ($a=10, b=20, c=91)$