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

arenceabigns 2021-08-22 Answered
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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Expert Answer

Obiajulu
Answered 2021-08-23 Author has 24220 answers
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.
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