# Discrete Math Question Provide a rule to define a relation R on N that is reflexive an

Discrete Math Question
Provide a rule to define a relation R on N that is reflexive and transitive but not symmetric. Prove that the relation you define is reflexive and transitive.

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Obiajulu
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.