Question

Give an example of a relation which is transitive $s$ symmetric $c$ but not reflexive.

Answer 1

In a set of integers, we can observe that the Relation

$R=\left\{\right(-1,-2),(-2,-3),(-2,-1),(-3,-1),(-1,-3),(-3,-2\left)\right\}$ does not satisfy the properties of a reflexive relation.

For example, if $a=b=1$, we can see that $(-1,-1)\notin R$. Hence it is not reflexive.

Given the relation $R=\left\{\right(-1,-2),(-2,-3),(-2,-1),(-3,-1),(-1,-3),(-3,-2\left)\right\}$, we can see that for any of the ordered pairs $(a,b)\in R\to (b,a)\in R$. Therefore $R$ is symmetric.

Given the Relation $R=\left\{\right(-1,-2),(-2,-3),(-2,-1),(-3,-1),(-1,-3),(-3,-2\left)\right\}$, we can observe that of any two pair of ordered pairs $(a,b),(b,c)\in R\to (a,c)\in R$.

Therefore, we can conclude that the relation $s$ transitive and symmetrical but not reflexive.