# Thread: Determine whether the following relation is transitive

1. ## Determine whether the following relation is transitive

Q) Determine whether the following relation is transitive
Relation R in the set N of natural numbers defined as
R = { (x,y) : y=x+5 and x<4}
The text book says that it is transitive. I dd not understand how it is transitive. Please help.

2. Originally Posted by myth31
Q) Determine whether the following relation is transitive
Relation R in the set N of natural numbers defined as
R = { (x,y) : y=x+5 and x<4}
The text book says that it is transitive. I dd not understand how it is transitive. Please help.
This is a trick question in that the relation $\displaystyle \mathcal{R}$ is vacuously transitive.
If $\displaystyle (a,b)\in \mathcal{R}$ then $\displaystyle b=a+5$ which implies that $\displaystyle b\ge 5$.
So that means that $\displaystyle \left( {\forall c \in \mathbb{N}} \right)\left[ {(b,c) \notin \mathcal{R}} \right]$
So the relation is transitive by default.