# Symmetric Relation question

• Nov 17th 2010, 05:03 PM
SunRiseAir
Symmetric Relation question
I know that a symmetric relation on a set X with for every a,b ∈X is noted by:

aRb ⇒ bRa

But I was wondering if this could be an alternative definition (please explain why it is not if this fails, thank you).

A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X

The reason I suspect this might be true is because the "relation" might be defined as an ordered pair. x is related to y by x and y matched as an ordered pair, (x, y).

• Nov 17th 2010, 05:06 PM
dwsmith
Quote:

Originally Posted by SunRiseAir
I know that a symmetric relation on a set X with for every a,b ∈X is noted by:
A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X

How does xRy and xRy imply yRx? Is there a typo here?

Having an extra xRy doesn't added anything so I am confused on what is going.

Since there is a Relation on X, $\displaystyle \displaystyle (x,y)\in X\rightarrow xRy$
• Nov 18th 2010, 01:08 AM
emakarov
Quote:

Originally Posted by SunRiseAir
A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X

You write $\displaystyle (x,y)\in X$. However, X is the set on which a relation (which one, by the way?) is considered, so there is no reason to assume that X contains pairs. To say that x and y are elements of X, one writes $\displaystyle x\in X$, $\displaystyle y\in X$ or $\displaystyle x,y\in X$. Some pedants even write $\displaystyle (x,y)\in X^2$.
• Nov 19th 2010, 06:54 PM
SunRiseAir
Quote:

Originally Posted by dwsmith
How does xRy and xRy imply yRx? Is there a typo here?

Having an extra xRy doesn't added anything so I am confused on what is going.

Since there is a Relation on X, $\displaystyle \displaystyle (x,y)\in X\rightarrow xRy$

What I mean is that for every x,y in X, xRy implies yRx.

This is the notation of a symmetric relation according to wikipedia.

http://en.wikipedia.org/wiki/Symmetric_relation

But then maybe that's a bad source.

Quote:

Originally Posted by emakarov
You write $\displaystyle (x,y)\in X$. However, X is the set on which a relation (which one, by the way?) is considered, so there is no reason to assume that X contains pairs. To say that x and y are elements of X, one writes $\displaystyle x\in X$, $\displaystyle y\in X$ or $\displaystyle x,y\in X$. Some pedants even write $\displaystyle (x,y)\in X^2$.

That's what I meant. I just forgot to right the square for X.

Let's say we're talking about a set whose elements are ordered pairs.

Could a symmetric relation in a set whose elements are ordered pairs be one that I described above?

Thanks again.
• Nov 19th 2010, 06:58 PM
dwsmith
Symmetry: when aRb, then bRa. Written as ordered pairs: $\displaystyle (a,b)\in R \ \mbox{and} \ (b,a)\in R$.
• Nov 20th 2010, 02:43 AM
emakarov
Quote:

A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X
This is like saying, "A subset of a given set X is called nonempty iff there exists an x in X". Since the left-hand side did not introduce a notation for the subset in question ("A subset Y of a given set X..."), the right-hand side cannot talk about this subset, and the whole definition is meaningless. Could you reformulate your definition and collect all relevant info in one post?