Well-definition

In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well-defined or ambiguous. A function is well-defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well-defined (and thus: not a function). The term well-defined is also used to indicate whether a logical statement is unambiguous.

A function that is not well-defined is not the same as a function that is undefined. For example, if f(x) = 1/x, then f(0) is undefined, but this has nothing to do with the question of whether f(x) = 1/x is well-defined. It is; 0 is simply not in the domain of the function.

Let $A_{0},A_{1}$ $A_{0},A_{1}$ be sets, let $A=A_{0}\cup A_{1}$ $A=A_{0}\cup A_{1}$ and "define" $f:A\rightarrow \{0,1\}$ $f:A\rightarrow \{0,1\}$ as $f(a)=0$ $f(a)=0$ if $a\in A_{0}$ $a\in A_{0}$ and $f(a)=1$ $f(a)=1$ if $a\in A_{1}$ $a\in A_{1}$ .

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