I was wondering, if a relation is reflexive, does that mean it is also antisymmetric?

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- Nov 2nd 2012, 12:46 PMBashyboyReflexive And Symmetric
I was wondering, if a relation is reflexive, does that mean it is also antisymmetric?

- Nov 2nd 2012, 01:10 PMPlatoRe: Reflexive And Symmetric
- Nov 2nd 2012, 02:01 PMBashyboyRe: Reflexive And Symmetric
It's neither, right?

- Nov 2nd 2012, 02:17 PMPlatoRe: Reflexive And Symmetric
- Nov 2nd 2012, 02:56 PMBashyboyRe: Reflexive And Symmetric
But doesn't the two elements (1, 2) and (2, 1) make it not reflexive?

- Nov 2nd 2012, 03:24 PMBashyboyRe: Reflexive And Symmetric
According to my textbook, "A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A." In the relation you gave, not every element a is related to itself.

- Nov 2nd 2012, 03:32 PMPlatoRe: Reflexive And Symmetric
Do you realize why you are having difficulty with this?

**You have absolutely no idea what any of this is about.**

Given any non-empty set $\displaystyle A$ if $\displaystyle \mathcal{R}\subseteq(A\times A)$

then $\displaystyle \mathcal{R}$ is a relation on $\displaystyle A$.

The diagonal is $\displaystyle \Delta_A=\{(x,x):~x\in A\}$.

If $\displaystyle \Delta_A\subseteq\mathcal{R}$ then $\displaystyle \mathcal{R}$**is reflexive**PERIOD!

Go back to reply #2. Does that relation contain the diagonal?

If it does then the relation is reflexive regardless of what else it contains.

If $\displaystyle \mathcal{R}=\mathcal{R}^{-1}$ then it is symmetric regardless. PERIOD! - Nov 2nd 2012, 03:34 PMBashyboyRe: Reflexive And Symmetric
I don't know why I began using this forum again. This is precisely why I ceased using this forum in the first place: because people have such rude tones in their writing.

- Nov 2nd 2012, 03:36 PMPlatoRe: Reflexive And Symmetric
- Nov 2nd 2012, 04:37 PMPlatoRe: Reflexive And Symmetric
- Nov 3rd 2012, 08:57 AMHartlwRe: Reflexive And Symmetric
The following is loosely copied from: http://www.cs.pitt.edu/~milos/course...es/Class21.pdf

Def: R on S1 S2 is a binary association between elements of S1 and S2.

Ex: S1={1,2,3}, S2={A,B,C}. R=(1,A) (2,3) (2,C)

Def: R on S1 is a binary association between elements of S1.

Ex: S1={1,2,3}. R=(1,2) ( 1,1) (3,3)

Def: R on A is reflexive if (a,a) ε R for__every element__of A.

Def: R on A is symmetric if (a,b) ε R → (b,a) ε R

Def: R on A is antisymmetric if [(a,b) ε R and (b,a) ε R] → a=b

If R is reflexive it might not be antisymmetric. (a,b) and (b,a) may belong to R but a unequal b - Nov 3rd 2012, 09:34 AMHartlwRe: Reflexive And Symmetric
- Nov 7th 2012, 11:49 AMHartlwRe: Reflexive And Symmetric
- Nov 7th 2012, 12:00 PMPlatoRe: Reflexive And Symmetric
- Nov 7th 2012, 12:10 PMHartlwRe: Reflexive And Symmetric
The possible relations on {1.2.3} are:

{(1,1), (2,2), (3,3)}

{(1,2), (2,3), (1,3)}

{(2,1) (3,2), (3,1)}

{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)}

{(1,1), (2,2), (3,3), (2,1), (3,2), (3,1)}

Unless you consider 1,2,3 to be symbols.

Sure, I can DEFINE things like 1=2, but they are meaningless for the integers.