Editing Well-defined expression (section)

=="Definition" as anticipation of definition==
In order to avoid the quotation marks around "define" in the previous simple example, the "definition" of <math>f</math> could be broken down into two logical steps:
{{ordered list
| ''The definition'' of the [[binary relation]]. In the example:
:<math>f := \bigl\{(a,i) \mid i \in \{0,1\} \wedge a \in A_i \bigr\}, </math>
(which so far is nothing but a certain subset of the [[Cartesian product]] <math>A \times \{0,1\}</math>.)
| ''The assertion''. The binary relation <math>f</math> is a function; in the example:
:<math>f: A \rightarrow \{0,1\}.</math>
}}
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 proven. That is, <math>f</math> is a function if and only if <math>A_0 \cap A_1 = \emptyset</math>, in which case <math>f</math> – as a function – is well defined.

On the other hand, if <math>A_0 \cap A_1 \neq \emptyset</math>, then for an <math>a \in A_0 \cap A_1</math>, we would have that <math>(a,0) \in f</math> ''and'' <math>(a,1) \in f</math>, which makes the binary relation <math>f</math> not ''functional'' (as defined in [[Binary relation#Special types of binary relations]]) and thus not well defined as a function. Colloquially, the "function" <math>f</math> is also called ambiguous at point <math>a</math> (although there is ''per definitionem'' never an "ambiguous function"), and the original "definition" is pointless. <br> <br />

Despite these subtle logical problems, it is quite common to use the term definition (without apostrophes) for "definitions" of this kind, for three reasons:

# It provides a handy shorthand of the two-step approach.
# The relevant mathematical reasoning (i.e., step 2) is the same in both cases.
# In mathematical texts, the assertion is "up to 100%" true.