# Question on how to tell if its reflective, symmetric, antisymmetric, & transitive?

• Apr 22nd 2009, 09:10 AM
Grillakis
Question on how to tell if its reflective, symmetric, antisymmetric, & transitive?
If I have t his:

x + y = 0

How do I tell if its: Reflective, Symmetric, Antisymmetric, and Transitive?

For Reflective, it says: check f(x,x) = 0

For symmetry, it says: check f(x,y) = f(y,x)

For Transitive, it says: check f(x,y)=0 and f(y,z)=0, then f(x,z)=0

What is Antisymmetric?

This is where I am getting this at: Is the x^2 - xy +2x -2y= 0 reflective/symmetric/transitive??? - Yahoo! Answers
(Look at User JCS)
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Am I doing this correct?

f(x, x) = x + y
= x + x
Reflective

f(y, x ) = x + y
= y + x
Symmetry

I am not sure on transitive and i do not know antisymmetric?
• Apr 22nd 2009, 10:59 AM
clic-clac
Hi

When you define a relation, a good thing to do is to define its domain ;)

As you write $x,y$ for the elements, maybe is it $\mathbb{R}$ ?

If so,

$x\mathcal{R}y \Leftrightarrow x+y=0
$

The relation $\mathcal{R}$ is reflexive if $\forall x\in\mathbb{R},\ x\mathcal{R}x$ i.e. $x+x=0.$ Is this true for any real $x$ ? ( A "no" means $\mathcal{R}$ isn't reflexive)

Using the same notation, $\mathcal{R}$ is:

symmetric if $\forall x\forall y(x\mathcal{R}y\Rightarrow y\mathcal{R}x)$

antisymmetric if $\forall x\forall y(x\mathcal{R}y\ \text{and}\ y\mathcal{R}x)\Rightarrow x=y$

transitive if $\forall x\forall y\forall z(x\mathcal{R}y\ \text{and}\ y\mathcal{R}z)\Rightarrow x\mathcal{R}z$