well-defined (original) (raw)

A mathematical concept is well-defined (German wohldefiniert, French bien défini), if its contents is on the form or the alternative representative which is used for defining it.

For example, in defining the http://planetmath.org/FractionPowerpower xr with x a positive real and r a rational number, we can freely choose the fraction form mn (m∈ℤ, n∈ℤ+) of r and take

and be sure that the value of xr does not depend on that choice (this is justified in the entry fraction power). So, the xr is well-defined.

In many instances well-defined is a synonym for the formal definition of a function between sets. For example, the function f⁢(x):=x2 is a well-defined function from the real numbers to the real numbers because every input, x, is assigned to precisely one output, x2. However, f⁢(x):=±x is not well-defined in that one input x can be assigned any one of two possible outputs, x or -x.

More subtle examples include expressions such as

Certainly every input has an output, for instance, f⁢(1/2)=3. However, the expression is _not_well-defined since 1/2=2/4 yet f⁢(1/2)=3 while f⁢(2/4)=6 and 3≠6.