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I'm reading the book Analysis 1 by Terence Tao. There is a concept of well-defined which is mentioned a lot of time in the book but I don't fully understand.

For example, in the picture below, in order to define the $subtraction$ $operation$ of 2 integers, he firstly defined the $negation$ $operation$ of integers.
After the definition, he stated that the definition is well-defined and we can prove that it is well-defined.

Could you please explain me what well-defined means and how can we prove something is well-defined or not ? Is there any collection of criteria for something to be well-defined ? ("something" here can be a definition, an operation)

Thank you very much for your help!

[Image] enter image description here

InTheSearchForKnowledge
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    Have you looked at Exercise 4.1.2? I would guess it might explain what well-defined means in this case. – DMcMor Jan 06 '21 at 16:23
  • Generally, when we partition a set (like $\mathbb N^2\to\mathbb Z$), and we then come up with operations on partitioned set by defining them on the original set, we need to make sure that the operations respect the partition, i.e., every element of the partition behaves identically under the operation, for it to make sense on the partitioned set. This is what well-definedness means in this context. – Rushabh Mehta Jan 06 '21 at 16:28
  • Without looking at the source, I'd say the issue here was that one can write an integer $n$ as $a-b$ in many ways. For example $3=4-1=10-7$. It is not instantly apparent that the definition of negation yields the same result regardless of which form you take. In my example, we'd have $-(4-1)=1-4=-3=7-10=-(10-7)$ so it all checks out. – lulu Jan 06 '21 at 16:29
  • For example, one could try to define a function on in the integers this way: given $n\in \mathbb Z$. write $n=a-b$ and define $F(n)=a$, but this would not be well defined. Sticking with the example in my previous comment, is $F(3)$ meant to be $4$ or $10$, or something else altogether? – lulu Jan 06 '21 at 16:31
  • I have just flaged this question, in want of better alternatives, as "needs more focus". It is unclear to me, if you are asking how to decide if something is "well defined", which is a rather hard question to answer, or if, much easier to answer, you are specifically asking about the additive inversion. Both questions would be valid, but are very, very different. – Gyro Gearloose Jan 06 '21 at 16:38
  • @GyroGearloose: Thanks for your response. So to be specific, I would like to know the definition of "well-defined" in 2 case. #1: Generally, meaning of well-defined for a definition #2: meaning of well-defined for an operation. Please enlighten me if there is still something unambiguous. Thanks for your help! – InTheSearchForKnowledge Jan 06 '21 at 16:45
  • @DucThinhVu so these are really two, very different questions. Let's wait what the moderators think about that. – Gyro Gearloose Jan 06 '21 at 16:48
  • @DucThinhVu In group theory, there is an axiom that states that there has to be (for to be a group) a neutral element "0" such that for every other element "u", $0+u=u$. There is another axiom that for every element $u$ there has to be another element $\bar u$ such that $u+\bar u=0$. – Gyro Gearloose Jan 06 '21 at 16:54

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In order for $-(a\mathbin —b):=(b\mathbin —a)$ to be well-defined we must have $-(a\mathbin — b)=-(c\mathbin —d)$ whenever $a\mathbin—b=c\mathbin—d$.

In general, this terminology is used when defining operations, functions, maps on equivalence classes. It means that the result is the same when a different representative of the same equivalence class is chosen.

Here is an example of an operation that is not well-defined: For a rational number $\frac{p}{q}$ (where $p,q$ are integers and $q\neq 0$) we could try to define $$ T\left( \frac p q \right) := \frac {p+1}{q+1}. $$ However, we have $\frac 1 3 = \frac 2 6$ but $$ T\left( \frac 1 3 \right) = \frac 2 4 \neq \frac 3 7 = T\left(\frac 2 6\right). $$ Hence, the expression $T(\frac p q)$ is not well-defined.

Christoph
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