Use of the term pair vs ordered pair

[Werg22]

Use of the term "pair" vs "ordered pair"

Why is it that authors use the term "pair" and "ordered pair" interchangeably and, maybe I'm mistaken, a little imprecisely? For example, in listing the field axioms, the language "for every pair x and y" is usually used. However, surly the author means "for every ordered pair x and y", otherwise, there is no need for the axiom of commutativity (neither for addition nor multiplication). Just something that has been bothering me.

 

[CompuChip]

Actually, when I hear "pair" I have in mind something like: (a, b) which is ordered by default, i.e. (b, a) is something different. Presumably the authors you are generally referring to have the same "problem"?

 

[matt grime]

"x and y" isn't an ordered pair, (x,y) would be an ordered pair.

What is bothering you about the phrase

"for every pair x,y we have xy=yx"

How is this axiom of commutativity redundant?

 

[Werg22]

I was under the impression that "pair" denoted a set of two objects (as opposed to "ordered pair" which denotes a set of two object in which order is important), therefore the pair x and y would be the same as the pair y and x. Defining addition and multiplication as functions assigning a unique x+y and xy to the pair "x and y" and then stating x+y=y+x or xy=yx is redundant; "x and y" is the same as "y and x" by the definition of a pair (as opposed to ordered pair), therefore x+y=y+x and xy=yx are implied at the outset and do not need to be stated as axioms.

 

[matt grime]

The term pair just denotes two things that have labels x and y. This is just common usage of English, it is not some statement about a set with two elements. If it were it would also imply x=/=y as well. Given pair with *labels* x and y, we assert there is something denoted xy, again the labelling is important. If we change the role of labels, as you do, to get yx, it does not imply that yx=xy.