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Math Help - Proving Something Is "Well-Defined"

  1. #1
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    Proving Something Is "Well-Defined"

    No specific problem in question here, I just have a question about this.

    I never quite grasped the concept of showing that something is "well-defined." What exactly does this mean? What is the general technique for showing that something is "well-defined?"

    Sorry if this seems like a silly question, but this concept just wasn't ever fully explained to me.
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  2. #2
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    Well-defined normally means what you defined does not lead to any contradictions.

    For example, I can try to define a function f from real numbers to real numbers s.t.
    f(0) = 1
    f(x + y) = f(x) + f(y)

    but this is not well defined because
    f(0) = 1 and
    f(0) = f(0 + 0) = f(0) + f(0) = 1 + 1 = 2

    Leading to a contradiction.

    For functions proving it is well-defined means proving that each input maps to exactly one output.

    This is just a small example. Normally well-defined has other meanings depending on context.
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  3. #3
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    Ultimately, "well-defined" means, well, that the definition makes mathematical sense. For each object involved in a well-formed definition, it must be clear how this object is constructed or why it exists.

    In snowtea's example, f is "over-defined" because there are two different ways to calculate f(0). In my experience, the need to prove that something is well-defined most often arises when the definition refers to one object of a whole set without saying how this object is chosen. In this case, it is possible that the concept in question is over-defined and, if another object is chosen from the set, the defined concept would be different or a contradiction would arise.

    Consider, for example, a set A, an equivalence relation R on A and a function f : A -> A that respects R, i.e., for all x, y in A, x R y implies f(x) R f(y). (Such equivalence relation is called a congruence.) Then one can consider the quotient set A / R, consisting of equivalence classes. Each class has the form [a] for some a in A and consists of all b in A such that a R b.

    Finally, we can define f : A / R -> A / R as follows: f([a]) = [f(a)]. One has to prove that this is well-defined. Indeed, functions have to map a single argument into a single value. If X is an equivalence class and a, b are in X, then f(X) can be [f(a)] or [f(b)], so one has to show that [f(a)] = [f(b)].

    Another way to botch a definition is to refer to a non-existing object, such as the least rational number greater that \sqrt{2}.

    Some set-theoretic paradoxes use the fact that some term is not defined precisely. For example, Richard's paradox takes for granted that for there is a way to tell if a given English phrase describes a unique real number.
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