Which of the following relations is symmetric but neither reflexive nor transitive for a set A = {1, 2, 3}.

The correct option is: R = {(1, 2), (2, 1)}

Let us analyze why this relation is symmetric but neither reflexive nor transitive on the set $A=\{1,2,3\}$ by checking each property step-by-step.

1. Reflexivity:
A relation $R$ on a set $A$ is reflexive if for every element $a\in A$, the ordered pair $(a,a)\in R$.
For the set $A=\{1,2,3\}$, a reflexive relation must contain the pairs $(1,1)$, $(2,2)$, and $(3,3)$.
Since $(1,1)\notin R$, $(2,2)\notin R$, and $(3,3)\notin R$, the relation is not reflexive.

2. Symmetry:
A relation $R$ is symmetric if whenever $(a,b)\in R$, then $(b,a)\in R$.
Let's check the elements of $R=\left\{\right(1,2),(2,1\left)\right\}$:
- For the pair $(1,2)\in R$, its reverse pair $(2,1)$ is also in $R$.
- For the pair $(2,1)\in R$, its reverse pair $(1,2)$ is also in $R$.
Since this condition holds for all pairs in $R$, the relation is symmetric.

3. Transitivity:
A relation $R$ is transitive if whenever $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$.
Let's test this with the elements in our relation:
We have $(1,2)\in R$ and $(2,1)\in R$.
For the relation to be transitive, the pair $(1,1)$ must also be in $R$.
However, $(1,1)\notin R$.
Therefore, the relation is not transitive.

Conclusion:
The relation $R=\left\{\right(1,2),(2,1\left)\right\}$ is symmetric, but it is neither reflexive nor transitive.