On the road to the City of Skepticism, I had to pass through the Valley of Ambiguity.
What is a function? The notion of a function is fundamental to mathematics, but like many fundamental aspects of mathematics, its definition was not nailed down until relatively recently. Mathematicians were using functions without thinking all that hard about what functions were! Let's rectify that.
The definition that I prefer to use is the following:
Definition. A function from the set to the set is an unambiguous rule which assigns to each element of an element of .
By unambiguous, I mean that each has only one element of assigned to it. This may seem obvious, but in fact there are some interesting mathematical operations that turn out to be a little ambiguous, for example:
Definition. We say that is a square root of if .
It's not hard to see that both and are square roots of , so the phrase ``the square root of " is actually ambiguous. There are various ways to deal with such ambiguity: for example, we could just content ourselves with the symbol , representing the two-element set , but then the square root of a number isn't a number!. The most straightforward way is just to exclude it by assumption, as I've done with the definition of function I gave above.
To functionize the notion of square root, we usually do the following:
Definition. We say that is the principal square root of , and write , if and .
That is, if there are the two possible choices for square root, we decide always to take the positive one. This notion of ``the square root" is now unambiguous, i.e. a function.
Let's connect this idea back to our formalism with relations. It's clear that expresses a relationship between and , so we can hope that our work on relations will help us develop a good definition of function as a kind of relation. For a function from to , we will want to consider pairs , so our relation (let's be traditional and call it ) will be a subset of .
What would it mean for the rule to be ambiguous? There would be some with and , where and are distinct. That is, being ambiguous means
Activity. Write a denial of the above logical sentence. This is what it means for to be unambiguous.
Activity. If your denial only includes and , alter it using logical facts so that it contains a .