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.
This is the notation of a symmetric relation according to wikipedia.
But then maybe that's a bad source.
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?
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?A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X