Reflexive And Symmetric

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November 2nd 2012, 12:46 PM
Bashyboy

Reflexive And Symmetric

I was wondering, if a relation is reflexive, does that mean it is also antisymmetric?
November 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 $A=\{1,2,3\}$ and $\mathcal{R}=\{(1,1),~(2,2),~(3,3),~(1,2),~(2,1)\}$ .

Is $\mathcal{R}$ reflexive?

Is $\mathcal{R}$ antisymmetric?
November 2nd 2012, 02:01 PM
Bashyboy

Re: Reflexive And Symmetric

It's neither, right?
November 2nd 2012, 02:17 PM
Plato

Re: Reflexive And Symmetric

Quote:

Originally Posted by Bashyboy

It's neither, right?

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

Is it antisymmetric?
November 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?
November 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.
November 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 $A$ if $\mathcal{R}\subseteq(A\times A)$
then $\mathcal{R}$ is a relation on $A$ .

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

If $\Delta_A\subseteq\mathcal{R}$ then $\mathcal{R}$ 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 $\mathcal{R}=\mathcal{R}^{-1}$ then it is symmetric regardless. PERIOD!
November 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.
November 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 $\{(1,1),(2,2),(3,3)\}$ so every element is related to itself.
November 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?
November 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
November 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
November 7th 2012, 11:49 AM
Hartlw

Re: Reflexive And Symmetric

Quote:

Originally Posted by Plato

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

Is $\mathcal{R}$ reflexive?

Is $\mathcal{R}$ antisymmetric?

R is impossible, ie, it is not a relation on integers.
November 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.
November 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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