Is Relation S Reflexive, Symmetric, and Transitive?

[kingstar]

Homework Statement

a) Consider the relation S defiend on the set {t : t is a person} such that xSy holds exactly if
person x is taller than y. Determine if the relation S is reflexive, symmetric and transitive.
Is the relation S an equivalence relation?

Homework Equations

Recall that a relation R dened on a set A is reflexive if for all x 2 A xRx.
Recall that a relation R dened on a set A is symmetric if for all x 2 A and y 2 A the xRy implies
yRx.
Recall that a relation R ened on a set A is transitive if for all x; y; z in A, both xRy and yRz
holds, then xRz holds as well.
Finally recall that a relation R is an equivalence relation if its reflexive, symmetric and transitive.

The Attempt at a Solution

As far as i can see the set is not symmetric or reflexive but I'm not 100% on transitive...
It would be transitive if x > y and y > z then x > z holds...but we aren't given any information on y > z?

So would this set be transitive?

 

[STEMucator]

Yes the relationship is not symmetric or reflexive, but it IS transitive.

If $xRy$ holds, then $x > y$, if $yRz$ holds, then $y > z$. So person $x$ is taller than person $y$ and person $y$ is taller than person $z$.

It must be the case that person $x$ must be taller than person $z$.

 

[kingstar]

Yes the relationship is not symmetric or reflexive, but it IS transitive.

If $xRy$ holds, then $x > y$, if $yRz$ holds, then $y > z$. So person $x$ is taller than person $y$ and person $y$ is taller than person $z$.

It must be the case that person $x$ must be taller than person $z$.

Yes, i agree with what you said...but the question does not mention that y > z

 

[STEMucator]

Yes, i agree with what you said...but the question does not mention that y > z

According to your definition (with some slight corrections):

Recall that a relation $R$ defined on a set $A$ is transitive if $\forall x, y, z \in A$, if both $xRy$ and $yRz$ hold, then $xRz$ holds as well.

Notice I added the word 'if'. If you assume they hold, does the conclusion still hold?

 

[kingstar]

Yeah then it holds. The definition is provided as part of the information the exam question but i'll just assume it in the exam as well. Thanks