Well-defined

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. 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 just cannot be in the domain of the function. 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 ightarrow {0,1}} as f ( a ) = 0 {displaystyle f(a)=0} if a ∈ A 0 {displaystyle ain A_{0}} and f ( a ) = 1 {displaystyle f(a)=1} if a ∈ A 1 {displaystyle ain A_{1}} . Then f {displaystyle f} is well-defined if A 0 ∩ A 1 = ∅ {displaystyle A_{0}cap A_{1}=emptyset } . This is e. g. the case when A 0 := { 2 , 4 } , A 1 := { 3 , 5 } {displaystyle A_{0}:={2,4},A_{1}:={3,5}} (then f(a) happens to be mod ⁡ ( a , 2 ) {displaystyle operatorname {mod} (a,2)} ). If however A 0 ∩ A 1 ≠ ∅ {displaystyle A_{0}cap A_{1} eq emptyset } then f {displaystyle f} is not well-defined because f ( a ) {displaystyle f(a)} is 'ambiguous' for a ∈ A 0 ∩ A 1 {displaystyle ain A_{0}cap A_{1}} . This is e. g. the case when A 0 := { 2 } {displaystyle A_{0}:={2}} and A 1 := { 2 } {displaystyle A_{1}:={2}} . Indeed, A 0 ∩ A 1 = { 2 } ∋ 2 {displaystyle A_{0}cap A_{1}={2} i 2} and f(2) would have to be 0 as well as 1, which is impossible. Therefore, the latter f is not well-defined and thus not a function. In order to avoid the apostrophes around 'define' in the previous simple example, the 'definition' of f {displaystyle f} could be broken down into two simple logical steps: Whereas the definition in step 1. is formulated with the freedom of any definition and is certainly effective (without the need to classify it as 'well-defined'), the assertion in step 2. has to be proved: If and only if A 0 ∩ A 1 = ∅ {displaystyle A_{0}cap A_{1}=emptyset } , we get a function f {displaystyle f} , and the f {displaystyle f} of 'definition' is well-defined (as a function).On the other hand: if A 0 ∩ A 1 ≠ ∅ {displaystyle A_{0}cap A_{1} eq emptyset } then for an a ∈ A 0 ∩ A 1 {displaystyle ain A_{0}cap A_{1}} there is both, ( a , 0 ) ∈ f {displaystyle (a,0)in f} and ( a , 1 ) ∈ f {displaystyle (a,1)in f} , and the binary relation f {displaystyle f} is not functional as defined in Binary relation#Special types of binary relations and thus not well-defined (as a function). Colloquially, the 'function' f {displaystyle f} is called ambiguous at point a {displaystyle a} (although there is per definitionem never an 'ambiguous function'), and the original 'definition' is pointless.Despite these subtle logical problems, it is quite common to anticipatorily use the term definition (without apostrophes) for 'definitions' of this kind, firstly because it is sort of a short-hand of the two-step approach, secondly because the relevant mathematical reasoning (step 2.) is the same in both cases, and finally because in mathematical texts the assertion is «up to 100%» true. The question of well-definedness of a function classically arises when the defining equation of a function does not (only) refer to the arguments themselves, but (also) to elements of the arguments. This is sometimes unavoidable when the arguments are cosets and the equation refers to coset representatives.