Check whether the relation R defined in the set $\{1, 2,3, 4, 5, 6\}$ as R $= \{(a, b): b$ = $a + 1\}$ is reflexive, symmetric or transitive.

Given R $= \{(a, b) : b = a + 1 \}, a, b \in \{1, 2, 3, 4, 5, 6\}$

$\Rightarrow R = \{(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)\}$

(i) Reflexive

Consider, $a \in \{ 1,\;2,\;3,\;4,\;5,\;6\} \Rightarrow a = a + 1$ which is false.

Therefore, $(a,\;a) \notin R.$ Thus, R is not reflexive.

(ii) Symmetric

Let $a,\;b \in \{ 1,\;2,\;3,\;4,\;5,\;6\}$

Consider, (a, b)$\in$R b $=$ a + 1

and (b, a)$\in$R $\Rightarrow$ a $=$ b + 1 which is false.

Therefore, R is not symmetric.

(iii) Transitive

Let, $a,\;b,\;c \in \{ 1,\;2,\;3,\;4,\;5,\;6\}$

Consider $a,\;b,\;c \in R \Rightarrow \;b = a + 1,\;(b,\;c) \in R$

$\Rightarrow$ $c = b + 1$

$\Rightarrow$ $c = a + 2\;\; \Rightarrow \;(a,\;c)\not \in$

Therefore, R is not transitive.

Hence, R is neither reflexive nor symmetric nor transitive.