Check whether the relation R defined on the set

Solution:

Reflexivity:

Let $a$ be an arbitrary element of R. Then,

$a=a+1$ cannot be true for all $a \in A$

$\Rightarrow(a, a) \notin R$

So, $R$ is not reflexive on $A$.

Symmetry:

Let $(a, b) \in R$

$\Rightarrow b=a+1$

$\Rightarrow-a=-b+1$

$\Rightarrow a=b-1$

Thus, $(b, a) \notin R$

So, $R$ is not symmetric on $A$.

Transitivity:

Let $(1,2)$ and $(2,3) \in R$

$\Rightarrow 2=1+1$ and $32+1$ is true.

But $3 \neq 1+1$

$\Rightarrow(1,3) \notin R$

So, $R$ is not transitive on $A$.