How do I prove that a function is well defined?

How do you in general prove that a function is well-defined?

f:XY:xf(x)

I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the following two things:

  1. Every element in the domain maps to an element in the codomain:
    xXf(x)Y
  2. The same element in the domain maps to the same element in the codomain:
    x=yf(x)=f(y)

At the moment I’m trying to prove this function is well-defined: f:(Z/12Z)(Z/4Z):[x]12[x]4, but I’m more interested in the general procedure.

Answer

When we write f:XY we say three things:

  1. fX×Y.
  2. The domain of f is X.
  3. Whenever x,y1,x,y2f then y1=y2. In this case whenever x,yf we denote y by f(x).

So to say that something is well-defined is to say that all three things are true. If we know some of these we only need to verify the rest, for example if we know that f has the third property (so it is a function) we need to verify its domain is X and the range is a subset of Y. If we know those things we need to verify the third condition.

But, and that’s important, if we do not know that f satisfies the third condition we cannot write f(x) because that term assumes that there is a unique definition for that element of Y.

Attribution
Source : Link , Question Author : Kasper , Answer Author : Asaf Karagila

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