# Properties of Relations

• Nov 30th 2007, 08:47 PM
MMM88
Properties of Relations
Hello,

I just have a question concerning properties of relations.
I'm wondering, if we wanna check wheter a relation is symmetric or not, is it ok to consider identical pairs..??

You know like.. R = {(a,b) | a^2 = b^2}.. R is defined on the integers.
Now the way I'm thinking of this is like.. lets take a look at (1,2) & (2,1) obviously the square of 1 doesnt equal the square of 2 so its not symmetric. But can we instead, consider a pair like (1,1) or (2,2).. which in that case can make the relation symmetric...???

Another question... can a relation be not symmetric and not anti-symmetric at the same time ???

I hope somebody can clarify theses point.
Thanks for your help.
• Dec 1st 2007, 12:07 AM
red_dog
Let $\displaystyle R$ be a binary relation.
$\displaystyle R$ is symmetric if $\displaystyle aRb\Rightarrow bRa, \ \forall a,b$.

Now, about our relation:
If $\displaystyle aRb\Rightarrow a^2=b^2\Rightarrow b^2=a^2\Rightarrow bRa$
So, the relation is symmetric.
• Dec 1st 2007, 12:09 AM
CaptainBlack
Quote:

Originally Posted by MMM88
Hello,

I just have a question concerning properties of relations.
I'm wondering, if we wanna check wheter a relation is symmetric or not, is it ok to consider identical pairs..??

You know like.. R = {(a,b) | a^2 = b^2}.. R is defined on the integers.
Now the way I'm thinking of this is like.. lets take a look at (1,2) & (2,1) obviously the square of 1 doesnt equal the square of 2 so its not symmetric. But can we instead, consider a pair like (1,1) or (2,2).. which in that case can make the relation symmetric...???

You relation $\displaystyle R$ is symmetric because if $\displaystyle (a,b) \in R$, then so is $\displaystyle (b,a)$

A specific pair can show that a relation is not symmetric, but cannot show
that it is symmetric.

Also a relation is not defined on all pairs. In this case $\displaystyle (a,b) \in R$ if and only
if $\displaystyle a^2=b^2$, $\displaystyle R$ has nothing to say about any other pairs.

RonL
• Dec 1st 2007, 12:15 AM
CaptainBlack
Quote:

Originally Posted by MMM88

Another question... can a relation be not symmetric and not anti-symmetric at the same time ???

$\displaystyle R=((1,2),(2,1),(3,4))$

is not anti-symmetric because $\displaystyle (1,2) \in R$ as is $\displaystyle (2,1)$, also it is not symmetric
because $\displaystyle (3,4) \in R$ but $\displaystyle (4,3) \not\in R$

RonL
• Dec 1st 2007, 03:35 PM
MMM88
Awesome!!! I get it now.
Thanks for the help guys. (Handshake)