# Reflexive And Symmetric

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• Nov 2nd 2012, 12:46 PM
Bashyboy
Reflexive And Symmetric
I was wondering, if a relation is reflexive, does that mean it is also antisymmetric?
• Nov 2nd 2012, 01:10 PM
Plato
Re: Reflexive And Symmetric
Quote:

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

Let $\displaystyle A=\{1,2,3\}$ and $\displaystyle \mathcal{R}=\{(1,1),~(2,2),~(3,3),~(1,2),~(2,1)\}$.

Is $\displaystyle \mathcal{R}$ reflexive?

Is $\displaystyle \mathcal{R}$ antisymmetric?
• Nov 2nd 2012, 02:01 PM
Bashyboy
Re: Reflexive And Symmetric
It's neither, right?
• Nov 2nd 2012, 02:17 PM
Plato
Re: Reflexive And Symmetric
Quote:

Originally Posted by Bashyboy
It's neither, right?

$\displaystyle \mathcal{R}$ is both reflexive and symmetric.

Is it antisymmetric?
• Nov 2nd 2012, 02:56 PM
Bashyboy
Re: Reflexive And Symmetric
But doesn't the two elements (1, 2) and (2, 1) make it not reflexive?
• Nov 2nd 2012, 03:24 PM
Bashyboy
Re: 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 PM
Plato
Re: Reflexive And Symmetric
Quote:

Originally Posted by Bashyboy
But doesn't the two elements (1, 2) and (2, 1) make it not reflexive?

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 PM
Bashyboy
Re: 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 PM
Plato
Re: Reflexive And Symmetric
Quote:

Originally Posted by Bashyboy
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.

Are you sure you are reading this correctly?
How do you dare say that/
It contains $\displaystyle \{(1,1),(2,2),(3,3)\}$ so every element is related to itself.
• Nov 2nd 2012, 04:37 PM
Plato
Re: Reflexive And Symmetric
Quote:

Originally Posted by Bashyboy
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.

Do you think it is fair of you to call us rude because you have difficulty reading mathematical writing such as that in your own textbook?
• Nov 3rd 2012, 08:57 AM
Hartlw
Re: 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 AM
Hartlw
Re: Reflexive And Symmetric
Quote:

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

Perhaps this is more comfortable:
R is reflexive if aRa for all a.
R is antisymmetric if aRb and bRa implies a=b.

reflexive does not imply antisymmetric. You could have aRb and bRa but not a=b
• Nov 7th 2012, 11:49 AM
Hartlw
Re: Reflexive And Symmetric
Quote:

Originally Posted by Plato
Let $\displaystyle A=\{1,2,3\}$ and $\displaystyle \mathcal{R}=\{(1,1),~(2,2),~(3,3),~(1,2),~(2,1)\}$.

Is $\displaystyle \mathcal{R}$ reflexive?

Is $\displaystyle \mathcal{R}$ antisymmetric?

R is impossible, ie, it is not a relation on integers.
• Nov 7th 2012, 12:00 PM
Plato
Re: Reflexive And Symmetric
Quote:

Originally Posted by Hartlw
R is impossible, ie, it is not a relation on integers.

How do you know that?
You have no clue what a relation is.
• Nov 7th 2012, 12:10 PM
Hartlw
Re: Reflexive And Symmetric
Quote:

Originally Posted by Plato
How do you know that?
You have no clue what a relation is.

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.
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