Determine if the relation R = (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,3), (4,1) on the set A = 1, 2, 3, 4 is reflexive, symmetric, or transitive.

A relation $R$ on a set $A$ is reflexive if for every element $a \in A$, the pair $(a, a)$ is in $R$.
The set is $A = \{1, 2, 3, 4\}$. For $R$ to be reflexive, it must contain $(1,1), (2,2), (3,3), (4,4)$.
Given $R = \{(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,3), (4,1)\}$.
We observe that $(4,4)$ is not in $R$.
Therefore, $R$ is not reflexive.

A relation $R$ on a set $A$ is symmetric if for every pair $(a, b) \in R$, the pair $(b, a)$ is also in $R$.
Let's check the pairs in $R$:

• $(1,2) \in R$, and $(2,1) \in R$. This part is symmetric.
• $(1,3) \in R$. For $R$ to be symmetric, $(3,1)$ must also be in $R$. However, $(3,1)$ is not in $R$. Therefore, $R$ is not symmetric.

A relation $R$ on a set $A$ is transitive if for every $a, b, c \in A$, if $(a, b) \in R$ and $(b, c) \in R$, then $(a, c)$ must also be in $R$.
Let's check for a counterexample:

• Consider $(4,1) \in R$ and $(1,2) \in R$. For $R$ to be transitive, $(4,2)$ must be in $R$. However, $(4,2)$ is not in $R$. Therefore, $R$ is not transitive.

Based on the checks, the relation $R$ is not reflexive, not symmetric, and not transitive.

Comparing our findings with the given options:
(A) $R$ is not reflexive but symmetric - Incorrect (not symmetric)
(B) $R$ is reflexive but not transitive - Incorrect (not reflexive)
(C) $R$ is neither reflexive, nor symmetric, nor transitive - Correct
(D) Equivalence Relation - Incorrect (an equivalence relation must be reflexive, symmetric, and transitive)

The final answer is $\boxed{\text{C}}$