What Defines a Nonreflexive Relation in Set Theory?

[Math Amateur]

I am reading the book: "Discovering Modern Set Theory. I The Basics" (AMS) by Winfried Just and Martin Weese.

I am currently focused on Chapter 2: Partial Order Relations ...

I need some help with understanding the meaning of "nonreflexive relation"...

The section from J&W giving the various definitions of properties of binary relations is as follows:View attachment 7544Can someone please give the definition of a nonreflexive relation (as distinct form an irreflexive relation ... ) ...

Is a nonreflexive relation a relation where \(\displaystyle \langle x,x \rangle \in R\) for some \(\displaystyle x\) but where there exist \(\displaystyle y\) such that \(\displaystyle \langle y, y \rangle \notin R\) ... ... ?
Hope someone can help ...

Peter

 

[Pereskia]

Hi,

Nonreflexive means that the relation is not reflexive. That is there is an x such that $(x,x)\not\in S$.
Irreflective is when there is no x such that $(x,x)\in S$

Regards
David

 

[HallsofIvy]

In the example given, the relation S= {(Kathy, Pam), (Pam, Kathy), (John, Paul), (Paul, John), (Kathy, Kathy), (Pam, Pam), (John, John), (Paul, Paul)} is reflexive because "for every x in F, (x, x) is in S". Here, F is {Kathy, Pam, Paul, John} so to be reflexive S must contain each of (Kathy, Kathy), (Pam, Pam), (John, John), and (Paul, Paul). If we were to remove anyone of those, say remove (Pam, Pam) to get {(Kathy, Pam), (Pam, Kathy), (John, Paul), (Paul, John), (Kathy, Kathy), (John, John), (Paul, Paul)}, that would no longer be "reflexive".

Notice that this relation contains (Kathy, Pam) and (Pam, Kathy) as well as (John,Paul) and (Paul, John) so is "symmetric"- whenever a symmetric relation contains (x, y) it must also contain (y, x).

I will leave it to you to show that "whenever the relation contains (x, y) and (y, z) then it contains (x, z)". the "transitive property", so that, in fact, this is an "equivalence relation" which is made clear from the description of the relation as "(x, y) is in the relation if and only if x and y are the same gender".

 

[Math Amateur]

In the example given, the relation S= {(Kathy, Pam), (Pam, Kathy), (John, Paul), (Paul, John), (Kathy, Kathy), (Pam, Pam), (John, John), (Paul, Paul)} is reflexive because "for every x in F, (x, x) is in S". Here, F is {Kathy, Pam, Paul, John} so to be reflexive S must contain each of (Kathy, Kathy), (Pam, Pam), (John, John), and (Paul, Paul). If we were to remove anyone of those, say remove (Pam, Pam) to get {(Kathy, Pam), (Pam, Kathy), (John, Paul), (Paul, John), (Kathy, Kathy), (John, John), (Paul, Paul)}, that would no longer be "reflexive".

Notice that this relation contains (Kathy, Pam) and (Pam, Kathy) as well as (John,Paul) and (Paul, John) so is "symmetric"- whenever a symmetric relation contains (x, y) it must also contain (y, x).

I will leave it to you to show that "whenever the relation contains (x, y) and (y, z) then it contains (x, z)". the "transitive property", so that, in fact, this is an "equivalence relation" which is made clear from the description of the relation as "(x, y) is in the relation if and only if x and y are the same gender".

Thanks Pereskia and HallsofIvy... I appreciate your help ...

Sorry for the late response ... have had to contend with other urgent matters ...

Peter