A relation R in the set $A = \{1,2,3,4\}$ is given by $R = \{(1,1), (2,2), (1,2), (2,3), (3,4), (4,4), (1,3), (2,4), (1,4)\}$ is

Transitive

The correct answer is Option (3) → Transitive

Given: $A = \{1, 2, 3, 4\}$

Relation: $R = \{(1,1), (2,2), (1,2), (2,3), (3,4), (4,4), (1,3), (2,4), (1,4)\}$

Check reflexivity:

All $(a,a)$ for $a \in A$ must be in $R$

$(1,1), (2,2), (4,4)$ are in $R$

$(3,3)$ is missing ⟹ Not reflexive

Check symmetry:

If $(a,b) \in R$, then $(b,a)$ must also be in $R$

$(1,2) \in R$ but $(2,1) \notin R$ ⟹ Not symmetric

Check transitivity:

Check if $(a,b), (b,c) \in R \Rightarrow (a,c) \in R$

$(1,2), (2,3) \in R \Rightarrow (1,3) \in R$ ✅

$(1,3), (3,4) \in R \Rightarrow (1,4) \in R$ ✅

$(2,3), (3,4) \in R \Rightarrow (2,4) \in R$ ✅

No counterexamples ⟹ Transitive