What is your definition for the equality of functions?
Hey everyone - I'm having trouble proving a theorem that I don't think should be too hard to prove:
Theorem (Equality of Functions):
If f: A -> B and g: A -> B are functions, then f=g if and only if f(a) = g(a) for all a that are an element of A.
Any help would be greatly appreciated.
That sentence makes no sense. A theorem is something you prove. If you are to prove this theorem, then it must be that you have a different definition of function equality that you're trying to show is equivalent to the one in the theorem. Otherwise, you have a definition here, not a theorem; you don't prove definitions.I believe that "equality of functions" just refers to what is being said in the theorem - that two functions are equal iff f(a) = g(a).
I'm used to thinking of the equality of functions this way: two functions are equal if and only if their domains are the same, and their rules of association are the same. And this is precisely what your theorem is saying. Therefore, in order to help you out, I need to know how exactly you are defining function equality.
It might also be appropriate if you tell us exactly how you defined "function" in our class. that just might be all we need. but yes, if you have any alternative definition of equality of functions, you should state that as well. It shouldn't be difficult once we know what definitions we can use.
Sorry about that - this is how it is written in the book:
"Two functions are equal if they have the same domain, the same codomain, and "agree" on every element of the domain. More formally,
Theorem: If f: A -> B and g: A -> B are functions, then f=g if and only if f(a) = g(a) for all a that are an element of A.
(Hint: a function is a relation. Saying that two functions are equal is to say they are relations between the same pair of sets, and they include the same ordered pairs.)"
One more definition to work with:
Definition: let A and B be nonempty sets. A function f from set A to set B (denoted by f: A -> B) is a relation between A and B satisfying the following conditions:
1. For each a that is an element of A, there exists b that is an element of B such that (a,b) is an element of f, and
2. if (a,b) and (a,c) are in f, then b=c.
If a is an element of A, the unique element b that is an element of B for which (a,b) is an element of f is denoted by f(a)