Empty Relations: Domain, Range & Functionality

[luizgguidi]

What is an empty relation?
Can an empty relation be a function?
Is an empty relation one with the empty set as its domain or as its range or both?

THanks

 

[HallsofIvy]

Look at the definitions! (How many times have I said that?)

A "relation" on a set A is a subset of the Cartesian product AxA. Since the empty set is a subset of any set, yes, the empty set is a subset of AxA and so is a relation.

A "function", F, is a relation on AxA such that "if x is in A, there is not more than one pair (x,y) with first member x". Because that says "not more than" it includes none. Yes, that is a function.

Any relation that has the empty set as its domain MUST have the empty set as its range and also is the empty relation.

 

[Hurkyl]

A "function", F, is a relation on AxA such that "if x is in A, there is not more than one pair (x,y) with first member x".

Actually, it's "there is exactly one pair".

And we should ask the original poster what precisely he means by "range" -- there are at least two distinct ways of using it here.

 

[Tac-Tics]

What is an empty relation?
Can an empty relation be a function?
Is an empty relation one with the empty set as its domain or as its range or both?

It's not really useful to worry about trivial points like this. It's all a matter of definition, and there is no universal definition for "function" or "relation". You can use any definition you want, but once you choose one, you have to stick with it.

In my mind, at least, I wouldn't even consider functions to be a kind of relation anyway. They are often defined in terms of relations, but they are very different grammatically. If R is a relation, "x R y" is a sentence. It can be either true or false. If f is a function, "f(x)" is just a noun. It has no statement behind it.