Why is the following relation transitive, but not reflexive?

Question

In a practice paper for an exam there is the following relation:

$$ E = \{(1,1),(2,2),(3,3),(4,4)\}\ \text{ on the set }V = \{1,2,3,4,5\} $$

It would appear that because $(5,5)$ is not in $E$, that it would not be a reflexive relation.

What is unclear is how the relation is still transitive. By our definition for Transitivity, for all elements in set $V$, there would have to be $(a,b)$ and $(b,c)\ldots$ and therefore $(a,c)$ in the set $E$.

This works for element $1$ in $V$ because you have $(1,1)$ and $(1,1)$ and therefore $(1,1)$. I would have assumed however that because there is no tuple $(5,5)$ in $E$ the entire relation could not be transitive?

Hope this makes sense, thanks for any input offered.

Answer 1

Reflexive means for all $a \in V$ then $(a,a) \in E$. That fails because $5 \in V$ and $(5,5) \not \in V$.

Transitive says nothing about items of $V$ but only about items in $E$. The relation is transitive if every time you have an $(a,b) \in E$ and a $(b,c) \in E$ you must also have a $(a,c) \in E$ then the relation is transitive. If you never have an $(5,x)$ or $(y, 5) \in E$ that will in no way affect what you do have in $E$.

This particular relationship if you have $(a,b) \in E$ then $a \ne 5; b\ne 5;$ and $a = b$. So if $(a,b), (b,c) \in E$ then $a = b = c$ and $(a,c) = (a,a) = (a,b) \in E$. So the relationship is transitive. That $a,b,c$ can never be equal to $5$ is utterly irrelevant.

Answer 2

Definition: $R$ is transitive if and only if $\forall x,y (R(x,y)$ and $R(y,x))\Rightarrow R(x,z)$

Can you point to where exactly you think this definition fails? For $x=y=5$ the antecedent fails to hold (since $R(5,5)$ is false) and therefore the implication is true, since an implication with a false antecedent is always true.

Answer 3

Your definition of transitivity is unclear as stated, and you have misunderstood it, there is no requirements on the members of the underlying set, only on the elements of the relation.

Answer 4

Proof $E$ is transitive: if $E\left(a,\,b\right)\land E\left(b,\,c\right)$ then $a=b\in V\backslash \left\{ 5\right\}$ and similarly $b=c\in V\backslash \left\{ 5\right\}$, so $a=c\in V\backslash \left\{ 5\right\}$ and $E\left(a,\,c\right)$.