This concept is quite important, since if you want to do mathematics rigour
ously you can only use terms that you have prove
d to be well-defined. Ironically
this term is quite difficult to define
, since any attempt to do so is bound to appear silly and tautological
Here is my attempt:
A property is well-defined if its definition is sufficient to determine it uniquely.
Sounds obvious and uninformative, doesn't it? Perhaps some examples will help.
We might want to define a semi-squaring function S: Q -> Q, ie a function which takes a rational and returns a rational. We define S(x) by: "Write x as p/q, with p, q are integers. Then S(x) = p2/q".
This operation is not very meaningful, eg take x = 2. Then x = 2/1, so S(2) = 4. But we could also write x = 4/2, giving S(2) = 8. This shows that the definition given for S is not sufficient to determine S(x) uniquely. S is not well-defined, and writing S(x) is therefore meaningless.
A property we certainly want to define for rationals is the sum. For rationals x, y we define x + y as follows: "Write x = a/b, y = c/d, where a, b, c, d are integers. Then x + y = (ad + bc)/bd". (This definition is in terms of addition and multiplication for integers.)
This seems perfectly reasonable, but before we can use the term 'sum' for rationals we have to make sure that the property is indeed well-defined.
Suppose that a/b = a'/b', c/d = c'/d'. Then
(a'd' + b'c')*bd = a'bd'd + b'bc'd = ab'd'd + b'bcd' = (ad + bc)*b'd' => (a'd' + b'c')/b'd' = (ad + bc)/bd
This shows that the sum of two rationals is independent of which representation in terms of integers we choose. Therefore our definition of the sum of two rationals is well-defined (provided that we have made sure that the sum and product of integers is well-defined). So we can happily add halves and quarters without worrying that our doing so will accidentally prove that 0=1.