Prove that $\{(a, a), (b, b), (c, c), (a, b), (b, a), (c, a), (a, c)\}$ is reflexive and symmetric but not transitive.

Question

Question: Let $A=\{a, b, c\}$ and let $R=\{(a, a), (b, b), (c, c), (a, b), (b, a), (c, a), (a, c)\}$. Prove that $R$ is reflexive and symmetric but not transitive.

I checked that the relation $R$ is reflexive and symmetric according to the following definition, but it's also transitive since $(a, a)∈R∧(b, b)∈R⇒(a, b)∈R$ and likewise transitive, for all elements.

By definition, $R$, a relation in a set $X$, is reflexive if and only if $\forall x\in X$, $x\,R\,x$.

$R$ is symmetric if and only if $\forall x, y\in X$, $x\,R\,y\implies y\,R\,x$.

R is transitive if and only if $\forall x, y, z\in X$, $xRy∧yRz⇒xRz$.

Answer 1

1. $R$ is reflexive because for $a,b,c\in A$, the elements $(a,a),(b,b),(c,c)\in R$. That is, $aRa, bRb, cRc$.
2. $R$ is symmetric because $R$ is reflexive, and $(b,a), (c,a)\in R$ whenever $(a,b), (a,c) \in R$, respectively. That is, if $(a,b),(a,c)\in R$ then $(b,a),(c,a)\in R$.
3. $R$ is not transitive because, for instance, $(c,a),(a,b)\in R$ but $(c,b)\notin R$.

Answer 2

There is $(b,a)$ , there is $(a,c)$ . For transitivity there should have been $(b,c)$ , but there is not.