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

Given set $A = \{1, 2, 3, 4\}$ and relation $R = \{(1,1), (2,2), (1,2), (2,3), (3,4), (4,4), (1,3), (2,4), (1,4)\}$

---

Checking if $R$ is reflexive:

For reflexive, every element must be related to itself: $(1,1), (2,2), (3,3), (4,4)$ must all be in $R$.

$(1,1)$ → Present

$(2,2)$ → Present

$(3,3)$ → Missing

$(4,4)$ → Present

$R$ is not reflexive because $(3,3)$ is missing.

---

Checking if $R$ is symmetric:

For symmetric, if $(a,b)$ is in $R$, then $(b,a)$ must also be in $R$.

$(1,2)$ is in $R$

For symmetric, $(2,1)$ must be in $R$

$(2,1)$ is not present.

$R$ is not symmetric.

---

Checking if $R$ is transitive:

For transitive, if $(a,b)$ is in $R$ and $(b,c)$ is in $R$, then $(a,c)$ must be in $R$.

$(1,2)$ and $(2,3)$ are in $R$ → $(1,3)$ is present ✓

$(1,2)$ and $(2,4)$ are in $R$ → $(1,4)$ is present ✓

$(2,3)$ and $(3,4)$ are in $R$ → $(2,4)$ is present ✓

$(1,3)$ and $(3,4)$ are in $R$ → $(1,4)$ is present ✓

All required pairs are present.

$R$ is transitive.

---

Therefore, the relation $R$ is transitive.