Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as

Solution:

Let A = {1, 2, 3, 4, 5, 6}.

A relation R is defined on set A as:

R = {(a, b): b = a + 1}

∴R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}

We can find $(a, a) \notin R$, where $a \in A$.

For instance,

(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∉ R

∴R is not reflexive.

It can be observed that $(1,2) \in R$, but $(2,1) \notin R$.

∴R is not symmetric..

 

Now, $(1,2),(2,3) \in \mathbf{R}$

But,

$(1,3) \notin R$

∴R is not transitive

Hence, R is neither reflexive, nor symmetric, nor transitive.