Step 1

A relation R on N is said to be

(i) Reflexive if \(\displaystyle\forall{a}\in{N},{a}{R}{a}\)

(ii) Symmetric if \(\displaystyle\forall{a}{R}{b},{a}{R}{b}\Rightarrow{b}{R}{a}\) and

(iii) Transitive if \(\displaystyle\forall{a}{R}{b}\) and \(\displaystyle{b}{R}{c}\Rightarrow{a}{R}{c}\)

Step 2

Now consider a relation \(\displaystyle{R}'\leq'\) on N

Now relation \(\displaystyle\leq\) is

(i) Reflexive: AS \(\displaystyle\forall{n}\in{N},{n}\leq{n}\)

(ii) Not symmetric: As \(\displaystyle{1}\leq{2}\) but 2 is not \(\displaystyle\leq{1}\)

(iii) Transitive: \(\displaystyle\forall{m}\leq{n}\) and \(\displaystyle{n}\leq{p}\Rightarrow{m}\leq{p}\)

Step 3

So the relation \(\displaystyle{R}'\leq'\) on N is reflexive, transitive but not symmetric.

A relation R on N is said to be

(i) Reflexive if \(\displaystyle\forall{a}\in{N},{a}{R}{a}\)

(ii) Symmetric if \(\displaystyle\forall{a}{R}{b},{a}{R}{b}\Rightarrow{b}{R}{a}\) and

(iii) Transitive if \(\displaystyle\forall{a}{R}{b}\) and \(\displaystyle{b}{R}{c}\Rightarrow{a}{R}{c}\)

Step 2

Now consider a relation \(\displaystyle{R}'\leq'\) on N

Now relation \(\displaystyle\leq\) is

(i) Reflexive: AS \(\displaystyle\forall{n}\in{N},{n}\leq{n}\)

(ii) Not symmetric: As \(\displaystyle{1}\leq{2}\) but 2 is not \(\displaystyle\leq{1}\)

(iii) Transitive: \(\displaystyle\forall{m}\leq{n}\) and \(\displaystyle{n}\leq{p}\Rightarrow{m}\leq{p}\)

Step 3

So the relation \(\displaystyle{R}'\leq'\) on N is reflexive, transitive but not symmetric.