What type of a relation is R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} on the set A – {1, 2, 3, 4}

The correct option is None of these.

Let us analyze the given relation $R$ defined on the set $A=\{1,2,3,4\}$ (note: the hyphen in "A – {1, 2, 3, 4}" in the question is a typo for equality, representing the set $A=\{1,2,3,4\}$).
The relation is given as:
$R=\left\{\right(1,3),(4,2),(2,4),(2,3),(3,1\left)\right\}$

We check the definition of each type of relation to see if it holds true for $R$:

1. Reflexive Relation:
A relation $R$ on set $A$ is reflexive if every element of $A$ is related to itself. That is, for all $a\in A$, $(a,a)\in R$.
For set $A=\{1,2,3,4\}$, the relation $R$ must contain $(1,1)$, $(2,2)$, $(3,3)$, and $(4,4)$ to be reflexive.
Since none of these elements (such as $(1,1)$) belong to $R$, the relation is not reflexive.

2. Symmetric Relation:
A relation $R$ on set $A$ is symmetric if whenever $(a,b)\in R$, then $(b,a)\in R$.
Let us check the pairs in $R$:
• For $(1,3)\in R$, we have $(3,1)\in R$.
• For $(4,2)\in R$, we have $(2,4)\in R$.
• For $(2,3)\in R$, the symmetric pair $(3,2)$ is not in $R$.
Since $(2,3)\in R$ but $(3,2)\notin R$, the relation is not symmetric.

3. Transitive Relation:
A relation $R$ on set $A$ is transitive if whenever $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$.
Let us test this property with some ordered pairs in $R$:
• We have $(4,2)\in R$ and $(2,3)\in R$.
For $R$ to be transitive, the pair $(4,3)$ must be in $R$.
Looking at the relation, $(4,3)\notin R$.
Since $(4,2)\in R$ and $(2,3)\in R$ but $(4,3)\notin R$, the relation is not transitive.

Therefore, since the relation is neither reflexive, symmetric, nor transitive, the correct option is "None of these".