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Well-defined and well-formed seem like pretty much the same concepts. I did read through the Wikipedia pages on both subjects. Can you elaborate on how they are different and provide examples (on a beginner level)?

Thanks.

Andrés E. Caicedo
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Emi Matro
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1 Answers1

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Well-defined is a property of functions; well-formed is a property of logical propositions. A well-defined function is any function that is single-valued (see Tim Gowers's site for a good explanation) and a well-formed proposition is any proposition that makes sense syntactically. This function is not well defined: $$ f(a/b)=a+b. $$ This proposition is not well-formed: $$ PQ\land. $$ (EDIT: It actually is well-formed if you use reverse Polish notation. But the string $(P\implies(QQ))$ is definitely not well-formed) Does that clear up the matter?

user134824
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  • sigh... PQ∧ is actually well-formed! It's just in reverse Polish notation http://en.wikipedia.org/wiki/Reverse_Polish_notation That said PQ∧R is not well-formed given that ∧ is a binary operation. – Doug Spoonwood Mar 21 '14 at 04:03
  • Okay, fair enough: I forgot about reverse Polish notation. Let me make something worse. – user134824 Mar 21 '14 at 04:09
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    I would add that “single-valued” is not always easy to determine. Consider a function $f:[0,1)\to{0,1,2,3,4,5,6,7,8,9}$ with the following definition: $f(x)$=$d_2$ if $x=0.d_1d_2d_3\dots$. (Here, $0.d_1d_2d_3$ is the decimal fraction with digits $d_1, d_2, \dots$.) This seems single-valued, because it seems to tell you exactly what $f(x)$ is. But real numbers don’t all have unique decimal fraction representations: $\frac{1}{10}=0.1000 \dots =0.099999\dots$, so does $f(\frac{1}{10})$ equal 0 or 9? In this case, $f$ isn’t a well-defined function. – Steve Kass Mar 21 '14 at 04:28
  • I like that example! – user134824 Mar 21 '14 at 04:35
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    As shown by the "counterexample" by Doug, well-formed is realtive to a set of formation rules : thus, you must specify an alphabet (set of synbols admissible), and then formation rules, defining the set of expressions (and not propositions) admissible. – Mauro ALLEGRANZA Mar 21 '14 at 06:52
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    Well defined is also a property of sets: a set is well defined if it is possible to tell whether a given object is, or is not, an element of that set. – Confutus Mar 21 '14 at 17:27