R3 = {(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4), (3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)} Determine whether the relation R3 is reflexive, symmetric, anti-symmetric and transitive. Determine whether the relation R3 is an equivalence relation or partial order. Give reason for your answer

Consider the relation $R_3 = \{\ (1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4), \\ (3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)\ \}$on the set $A=\{1,2,3,4\}.$

1. Since $(a,a)\in R_3$ for each $a\in A,$ the relation $R_3$ is reflexive. Taking into account that $(a,b)\in R_3$ implies $(b,a)\in R_3$ for any pair $(a,b)\in R_3,$ we conclude that the relation $R_3$ is symmetric. Since $(1,2)\in R_3$ and $(2,1)\in R_3,$ this relation is not anti-symmetric. Taking into account that $(a,b)\in R_3$ and $(b,c)\in R_3$ implies $(a,c)\in R_3$ for any pairs $(a,b),(b,c)\in R_3,$ we conclude that the relation $R_3$ is transitive.
2. Since the relation $R_3$ is reflexive, symmetric and transitive, it is an equivalence relation. Taking into account that $R_3$ is not anti-symmetric, we conclude that $R_3$ is not a partial order.