Using logic symbols "fluently" in mathematics. How can I show that one set of conditions, represented by logic symbols, is independent of another?

Question

I am attempting to improve my skills in proof-writing. In doing so, many of the mathematical statements that I encounter involve many "sets" of "independent" conditions, which are conventionally represented using logic symbols.

I want to give fair warning to readers: I am a novice to logic and mathematical proof-writing, and as such, some of what follows may seem like gibberish. This is likely due to my stated inexperience. However, I want to assure readers that it is not my intention to waste their time. My goal is to learn to form precise and elegant mathematical statements using the conventional symbolism.

For instance, take the mathematical statement, "Show that the real number is only divisible by 1 and itself". When writing this mathematically, I want to convert the mathematical english precisely into mathematical/logical symbols/notation. However, I want the purely mathematical representation of the statement to be "fluent". This "fluency" depends on the "independence" of sections of the mathematical statements -- represented in english by periods (.) or commas (,).

Show that $\dfrac{n}{k} = c$

This statement is obviously incomplete; it requires further information, which is usually added using mathematical/logical notation:

(1) $\forall \ n \in \mathbb{Z} > 1$

(2) $k = 1 \lor k = n$

Notice this mathematical statement requires two "independent" conditions for it to make sense? When I say independent, I mean that one condition is independent from the other.

And other mathematical statements could have more than $2$ independent conditions.

If we were to simply join these together, we would get something that isn't a fluent, precise, or a correct representation of the mathematical english:

$\dfrac{n}{k} = c \ \forall \ n \in \mathbb{Z} > 1 \ k = 1 \lor k = n$

Notice also how the logic symbols "overlap", since there is no equivalent of a period (.) or comma (,). In other words, there is no way to distinguish which sections of the mathematical statement are independent or dependent.

So how do I join the mathematical statement $\dfrac{n}{k} = c$ with the conditions $\forall \ n \in \mathbb{Z} > 1$ and $k = 1 \lor k = n$? In other words, how do I convert the mathematical english to a fluent and correct representation using mathematical/logical notation?

This is something that has been troubling me for quite a while, and I'm eager to learn the correct way to write these. I hope my question has been able to convey this effectively.

Thank you.

Answer 1

Barring formal proof in the study of logic, it is advised to use full sentences as opposed to symbols in good mathematical writing. That is, instead of saying:

$ \def\zz{\mathbb{Z}} $

We have $$\forall n,k \in \zz\ \left ( n > 1 \land ( k = 1 \lor k = n ) \to \exists c \in \zz\ \left ( \frac{n}{k} = c \right) \right ) $$

You would say:

Consider an arbitrary $n$, where $n\in \mathbb{Z}$ and $n > 1$, and let $k = 1$ or $k = n$ . Show that: $$ \frac{n}{k} = c $$

Unless you need to write in first order logic (for proof checkers or Fitch style derivations), the latter method is orders of magnitude more readable. Good writing involves striking a balance between understandability and brevity/unambiguity. If you want to learn to write good proof in the usual sense, make your writing easier for the reader!

Here are a bunch of resources that will drill this. Every single one of these, in some sense, says what my answer here states: avoid this type of shorthand.

Source 1

Source 2

Source 3

Source 4

Answer 2

The right way to write your statement symbolically (according to my guess of what you mean) is: $ \def\zz{\mathbb{Z}} \def\lfrac#1#2{{\large\frac{#1}{#2}}} $

$\forall n,k \in \zz\ \Big( n > 1 \land ( k = 1 \lor k = n ) \to \exists c \in \zz\ \big( \lfrac{n}{k} = c \big) \Big)$

By the way, it is always possible to express any mathematical statement in symbolic form. In fact, for almost all of modern mathematics, if it cannot be done in some fixed format (such as first-order sentence over ZFC) then it is not a mathematical statement! However, as others have pointed out, in mathematical writing we use symbols to facilitate understanding, and not just for the sake of conciseness or precision. Even if the goal is for it to be computer-checkable, it is very uncommon for symbols as opposed to ASCII keywords to be used, because it is difficult to type symbols. For example Coq uses the keyword "forall" and not the symbol "$\forall$".

In my opinion, a clear way to express your statement in writing would be:

Given any integer $n > 1$, if $k = 1$ or $k = n$ then $\lfrac{n}{k}$ is an integer.

It is barely a little longer than the symbolic form, and yet conveys all the information in an easily readable English sentence. If you want a shorter version you can use more symbols (which would need more prior definitions):

Given any $n \in \zz_{>1}$, if $k \in \{1,n\}$ then $\lfrac{n}{k} \in \zz$.

It is clearly a trade-off between readability and conciseness. Note that a significant factor is the use of set-theoretic notation, which is commonly used in modern mathematical writing. Also, it is an understood convention to leave out universal quantifiers at the outermost level like for "$k$" here.

Freely using set-theoretic notation with first-order logic, we can write a short but clearly equivalent version of your statement:

$\forall n,k \in \zz\ \big( n > 1 \land k \in \{1,n\} \to \lfrac{n}{k} \in \zz \big)$.