Well-defined expression

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 not well defined, 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. For instance, if f {\displaystyle f} takes real numbers as input, and if f ( 0.5 ) {\displaystyle f(0.5)} does not equal f ( 1 / 2 ) {\displaystyle f(1/2)} then f {\displaystyle 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. A function that is not well defined is not the same as a function that is undefined. For example, if f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} , then even though f ( 0 ) {\displaystyle f(0)} is undefined, this does not mean that the function is not well defined; rather, 0 is not in the domain of f {\displaystyle f} . Let A 0 , A 1 {\displaystyle A_{0},A_{1}} be sets, let A = A 0 ∪ A 1 {\displaystyle A=A_{0}\cup A_{1}} and "define" f : A → { 0 , 1 } {\displaystyle f:A\rightarrow \{0,1\}} as f ( a ) = 0 {\displaystyle f(a)=0} if a ∈ A 0 {\displaystyle a\in A_{0}} and f ( a ) = 1 {\displaystyle f(a)=1} if a ∈ A 1 {\displaystyle a\in A_{1}} . Then f {\displaystyle f} is well defined if A 0 ∩ A 1 = ∅ {\displaystyle A_{0}\cap A_{1}=\emptyset \!} . For example, if A 0 := { 2 , 4 } {\displaystyle A_{0}:=\{2,4\}} and A 1 := { 3 , 5 } {\displaystyle A_{1}:=\{3,5\}} , then f ( a ) {\displaystyle f(a)} would be well defined and equal to mod ⁡ ( a , 2 ) {\displaystyle \operatorname {mod} (a,2)} .