A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$

Let's take any two numbers, say $\{1,2\}$. We can observe that the Relation $R = \{(1,2), (2,1)\}$ satisifes the properties of a symmetrical relation.

Here, $(1,2) \in R \rightarrow (2,1) \in R$. Hence it is symmetric.

Given $R = \{(1,2), (2,1)\}$, we can observe that if $a=1$, $(1,1) \not \in R$. Therefore, R is not reflexive.

Given $R = \{(1,2), (2,1)\}$ we can further observe that while $(a,b) = (1,2) \in R$ and $(b,c) = (2,1) \in R$, which implies that $a=1, b=2, c=1$, we see that $(a,c) = (1,1) \not \in R$. Hence $R$ is not transitive.

Therefore, given a set of integers for instance, the Relation $R = \{(1,2), (2,1)\}$ is the example of a relation that is symmetric, but not reflexive or transitive.