Can anyone please help me to understand what does "well-defined" mean in the definition of Set?

"Well-defined" means that the definition indeed specifies one and only one object.

For example

  • Let $n$ be the even prime.

    This makes $n$ well-defined, because there is exactly one even prime, $2$.

  • Let $n$ be the prime between $24$ and $28$.

    This looks like a definition but is not well-defined. There is no prime between $24$ and $28$.

  • Let $n$ be the prime below $10$.

    Again, this is not well-defined, this time because there are several primes below $10$. Note that by saying “the” you claim uniqueness.

  • Let $n$ be the smallest composite prime.

    Again, not well-defined. There is no composite prime because the two notions “composite” and “prime” contradict each other.


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 $$\tan\frac\pi2 := \lim_{x\to\frac\pi2}\tan x = \infty.$$ But you probably haven't done this.)

People say a set is "well-defined" to mean that there aren't any problems/contradictions/inconsistencies (like the above) when defining it.


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).

Tags:

Abstract Algebra

Elementary Set Theory

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