Can anyone please help me to understand what does well-defined mean? Let's say $X = \< 1 , 2 , 3 , \tan \frac<\pi> \>$ . Is $X$ a set ? $\tan x$ tends to infinity when $x \in (0 , \frac<\pi>)$ and $x$ tends to $\frac<\pi>$ . And $\tan x$ tends to minus infinity when $x \in ( \frac<\pi> , \pi )$ and $x$ tends to $\frac<\pi>$ . But we do not have any concrete idea about $\tan \frac<\pi>$ . So it is undefined. So $X$ can not be called a set. Am I correct ?
$\begingroup$ Where is this quote block from? It looks like an attempt to define a set, but I always learned that set is a primitive that isn't defined. Of course, you can always give sentences that try to convey the natural meaning of a concept, but then one needn't go digging too deeply about the terms used. $\endgroup$
Commented Jun 15, 2020 at 12:49"Well-defined" means that the definition indeed specifies one and only one object.
Is $X$ a set? I think it is not because $\tan\frac<\pi>2$ is infinity.
Guessing your context, you are correct. I would technically say that, since $\frac\pi2$ is not in the domain of $\tan$ , the object $\tan\frac\pi2$ is undefined.
(Unless, maybe if you have previously defined $\infty$ is as an object, and defined $\tan\frac\pi2$ to be $$\tan\frac\pi2 := \lim_
People say a set is "well-defined" to mean that there aren't any problems/contradictions/inconsistencies (like the above) when defining it.
answered Jun 13, 2020 at 4:56 1,680 7 7 silver badges 15 15 bronze badges $\begingroup$ One book said " 'Well defined' are not words of praise." $\endgroup$ Commented Jun 13, 2020 at 5:19 $\begingroup$ @DanielWainfleet what is that supposed to mean? $\endgroup$ Commented Jun 13, 2020 at 5:27$\begingroup$ I guess that it's a joke. Well followed by a past particple is often said in praise, e.g. "well done", "well played", so "well defined" sounds as if you are praising someone for doing a good job of defining something. $\endgroup$
Commented Jun 13, 2020 at 6:01 $\begingroup$ ahh, now i get it. thank you badjohn. $\endgroup$ Commented Jun 13, 2020 at 6:18$\begingroup$ @Zest. It was in the context of defining the operation * as (aH)*(bH)=abH when H is a normal sub-group of a group G, when a, b $\in$ G. The author then explained that it meant that if aH=a'H and bH=b'H then (aH)*(bH)=(a'H)*(b'H). $\endgroup$
Commented Jun 13, 2020 at 15:18 $\begingroup$The term "well-defined" is not being used to refer to the domain of definition of a partial function (like $\tan$ ) here, but rather to the fact that not every purported definition defines a set.
A famous example is Bertrand Russell's set of sets that do not contain themselves: $$ R = \ < x \mid x \not\in x \>$$ Then if $R \in R$ , this implies that $R \not\in R$ , while if $R \not\in R$ , unfortunately $R \in R$ . Either way we get a contradiction.
The way we use sets nowadays starts with certain sets (e.g. $\omega$ , the set of natural numbers) as given and defines others as subsets, and does not allow us to define $R$ , so we avoid this contradiction (we cannot prove that a contradiction is avoided, but that just a general feature of mathematical theories that can express enough arithmetical facts and for which the set of provable statements is computably enumerable, nothing to do with set theory in particular).
answered Jun 13, 2020 at 7:19 Robert Furber Robert Furber 1,409 12 12 silver badges 22 22 bronze badges $\begingroup$It is well to notice that what you quoted is not an actual definition of a set in axiomatic set theory where sets are undefined terms with certain axiomatic properties. It is similar to the original definitions of Cantor who founded set theory. For example, a quote from 1895
By a 'set' we understand every collection to a whole $M$ of definite, well-differentiated objects $m$ of our intuition or our thought. (We call these objects the 'elements' of $M$ .)
This is similar to dictionary "definitions" of words which use other word phrases in the definitions, but not everything can be defined this way. There must be first given a number of undefined words from which all other words are defined. For example, what exactly is a "collection"? The key concept turns out to be that of elementhood. That is, it must be always possible to be able to definitely decide if $m$ is an element of $M$ or is not, for any given $m$ and $M$ .