Well-defined Expression

Note Summary

A mathematical concept is well-defined if it yields a unique, consistent result regardless of representation. Otherwise, it’s not well-defined.

Definition

In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise the expression is said to be ill-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.

Easy human example, let $f$ but a function. If $f(0.5)\ne f(\frac{1}{2})$ then $f$ is not well defined (and thus not a function).

The term well-defined can also be used to indicate that a logical expression is unambiguous or uncontradictory.

Example

Let $A_0,A_1$ be sets, let $A=A_0\cup A_1$ and “define” $f:A\to \{0,1\}$ as $f(a)=0$ if $a\in A_0$ and $f(a)=1$ if $a\in A_1$.

Then $f$ is well-defined if $A_0\cap A_1=\varnothing$. For example, if $A_0:=\{2,4\}$ and $A_1:=\{3,5\}$, then $f(a)$ would be well-define and equal to $\mathrm{mod}(a,2)$.

However, if $A_0:=\{2,4\}$ and $A_1:=\{4,5\}$, then $f(4)$ would be ill-defined since $4\in A_0$ and $4\in A_1$, so $f(4)$ could be either $0$ or $1$.

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