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" 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" 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", then the fact that "f"(0) is undefined does not mean that the "f" is "not" well defined – but that 0 is simply not in the domain of "f". Example. Let formula_1 be sets, let formula_2 and "define" formula_3 as formula_4 if formula_5 and formula_6 if formula_7. Then formula_8 is well defined if formula_9. For example, if formula_10 and formula_11, then formula_12 would be well defined and equal to  formula_13. However, if formula_14, then formula_8 would not be well defined because formula_12 is "ambiguous" for formula_17. For example, if formula_18 and formula_19, then formula_20 would have to be both 0 and 1, which makes it ambiguous. As a result, the latter "formula_8" is not well defined and thus not a function. Section::::"Definition" as anticipation of definition. In order to avoid the quotation marks around "define" in the previous simple example, the "definition" of formula_8 could be broken down into two simple logical steps: While 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. That is, formula_8 is a function if and only if formula_24, in which case formula_8 – as a function – is well defined. On the other hand, if formula_14, then for an formula_17, we would have that formula_28 "and" formula_29, which makes the binary relation formula_8 not "functional" (as defined in Binary relation#Special types of binary relations) and thus not well defined as a function. Colloquially, the "function" formula_8 is also called ambiguous at point formula_32 (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 – for three reasons: Independence of representative. 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, serving as representatives. This is sometimes unavoidable when the arguments are cosets and the equation refers to coset representatives. The result of a function application must then not depend on the  choice of representative. Functions with one argument. For example, consider the following function where formula_34 and formula_35 are the integers modulo "m" and formula_36 denotes the congruence class of "n" mod "m". N.B.: formula_37 is a reference to the element formula_38, and formula_39 is the argument of "formula_8". The function "formula_8" is well defined, because As a counter example, the converse definition does not lead to a well defined function, since e.g. formula_44 equals formula_45 in formula_46, but the first would be mapped by formula_47 to formula_48, while the second would be mapped to formula_49, and formula_48 and formula_49 are  unequal in formula_52. Operations. In particular, the term "well defined" is used with respect to (binary) operations on cosets. In this case one can view the operation as a function of two variables and the property of being well defined is the same as that for a function. For example, addition on the integers modulo some "n" can be defined naturally in terms of integer addition. The fact that this is well defined follows from the fact that we can write any representative of formula_54 as formula_55, where formula_56 is an integer. Therefore, and similarly for any representative of formula_58 , thereby making formula_59 the same irrespective of the choice of representative. Well-defined notation. For real numbers, the product formula_60 is unambiguous because formula_61 (and hence the notation is said to be "well defined"). This property, also known as associativity of multiplication, guarantees that the result does not depend on the sequence of multiplications, so that a specification of the sequence can be omitted. The subtraction operation, on the other hand, is not associative. However, there is a convention that formula_62 is shorthand for formula_63, thus it is "well defined". Division is also non-associative. However, in the case of  formula_64, parenthesization conventions are not so well established, so this expression is often considered ill defined. Unlike with functions, the notational ambiguities can be overcome more or less easily by means of additional definitions (e.g., rules of precedence, associativity of the operator). For example, in the programming language C the operator codice_1 for subtraction is "left-to-right-associative", which means that codice_2 is defined as codice_3, and the operator codice_4 for assignment is "right-to-left-associative", which means that codice_5 is defined as codice_6. In the programming language APL there is only one rule: from right to left  – but parentheses first. Other uses of the term. A solution to a partial differential equation is said to be well defined if it is determined by the boundary conditions in a continuous way as the boundary conditions are changed.