Queries related to Transitive Relations

Question

Let me assume that 'R' is a relation on 'A'. 'A' contains four elements {x,y,z,o}.

Also let x R y , y R z and x R z . Now, my queries are :

Will 'R' be called a Transitive Relation if :

1) o is not related to any element in 'A' under relation 'R'

2) o is related only to itself under relation 'R'

3) o is related to another element , say y , under relation 'R'

My answers : 1) Yes 2) Yes. 3) No

Am I correct?

Answer 1

Assuming $x$R$y$, $y$R$z$, $xRz$ is still valid for all 3 questions, then you are correct.

3) having $o$R$y$ makes it not transitive because $o$ is not related to $z$

Answer 2

Actually the set A doesn't play vital role to define R is transitive, R is transitive if R contains (x,z), when ever it contains(x,y) and (y,z).

So using above argument the answers yes, yes, no are correct.

In third question, o is related y is given, and y is related to z is already know, but is o is related to z or not we don't know. So third question we can decide only when we the relation between o and z.