Checking relation for reflexive, symmetric and transitive

[issacnewton]

Homework Statement

Relation $R$ is defined on set $\mathbb{N}$ of natural numbers defined as
$$R = \big \{ (x,y) | y = x + 5 \text{ and } x < 4 \big \} $$

Relevant Equations

Definition of reflexivity, symmetry and transitivity

Now, with the given set of natural numbers, we can deduce the relation $R$ to be as following

$$ R = \big \{ (1,6), (2,7), (3,8) \big \} $$

Now, obviously this is not a reflexive and symmetric. And I can also see that this is transitive relation. We never have $(a,b) \in R$ and $(b,c) \in R$ since if $(a,b) \in R$, then we have $b \nless 4$ and so we can not have $(b,c) \in R$. So, in the definition of the transitivity, the antecedent is always false. And since the definition involves an implication, the implication will always be vacuously true. So, the relation is transitive . So, the relation $R$ is transitive but not reflexive or symmetric. Now, I have used the implication and truth table of implication to see that the relation is transitive. I have to teach this to some students and they have not covered logic and implication. They have only studied basics about the methods of proof like, direct proof, proof by contradiction, proof by contra positive etc. So, how would I explain this to these students ?

Thanks

 

[member 587159]

If there are no elements for which a statement might hold, then clearly all those elements satisfy the statement (as there are none!).

Not sure what's more to say about it than that. Are you looking for an intuitive answer? Or real life analogies?

 

[issacnewton]

I think your explanation might suffice for these students I will be teaching. Thanks