Thread: 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).




Thanks for your time.

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,

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 . 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 , or . Some pedants even write .

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,

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.

Originally Posted by emakarov

You write . 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 , or . Some pedants even write .

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.

Symmetry: when aRb, then bRa. Written as ordered pairs: .

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?