I was wondering, if a relation is reflexive, does that mean it is also antisymmetric?
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I was wondering, if a relation is reflexive, does that mean it is also antisymmetric?
Let $\displaystyle A=\{1,2,3\}$ and $\displaystyle \mathcal{R}=\{(1,1),~(2,2),~(3,3),~(1,2),~(2,1)\}$.Quote:
Originally Posted by Bashyboy
Is $\displaystyle \mathcal{R}$ reflexive?
Is $\displaystyle \mathcal{R}$ antisymmetric?
It's neither, right?
$\displaystyle \mathcal{R}$ is both reflexive and symmetric.Quote:
Originally Posted by Bashyboy
Is it antisymmetric?
But doesn't the two elements (1, 2) and (2, 1) make it not reflexive?
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.
Do you realize why you are having difficulty with this?Quote:
Originally Posted by Bashyboy
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!
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.
Are you sure you are reading this correctly?Quote:
Originally Posted by Bashyboy
How do you dare say that/
It contains $\displaystyle \{(1,1),(2,2),(3,3)\}$ so every element is related to itself.
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?Quote:
Originally Posted by Bashyboy
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
Perhaps this is more comfortable:Quote:
Originally Posted by Bashyboy
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
R is impossible, ie, it is not a relation on integers.Quote:
Originally Posted by Plato
How do you know that?Quote:
Originally Posted by Hartlw
You have no clue what a relation is.
The possible relations on {1.2.3} are:Quote:
Originally Posted by Plato
{(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.
