How can I convert $A =$ {$c > 1$ : there exists a natural number $m$ ... to mathematical notation?

Question

In this post A = {c > 1 : there exists a natural number m, such that for every n > m, there is a prime between n and cn}.

Now I don't want to discuss on the problem itself, but only on how to write A = {c > 1 : there exists a natural number m, such that for every n > m, there is a prime between n and cn} using mathematics. Now here's what I wrote:

Let $P$ be a propriety of a number $c$; $P(c)=\exists m \in \mathbb{N} : \forall n>m, n \in \mathbb{N} \rightarrow \exists p: p\ \text{is prime}, cn>p>n$. Now let $A=\{c>1 \ | \ P(c)\}$

Is this correct? If it's wrong how can I write it?

Answer 1

Using fairly standard notation, $P(c)$ could be described as follows.

$$\exists m \in \mathbb{N}\;\; \forall n>m\;\; \exists p \in \mathbb{N} \,:\, \mathrm{isPrime}(p) \,\&\, (cn > p > n)$$

You see a lot of variants on this basic pattern; for example, some people might write

$$\exists m \in \mathbb{N}, \forall n>m, \exists p \in \mathbb{N}, \,\mathrm{isPrime}(p) \wedge (cn > p > n)$$

However, people generally use words for such things, since experience has taught us that definitions are often easier to understand when they're phrased in English, rather than "Symbolish." So following quid's suggestion, one might write:

Let $A$ be the set of all $c∈ℝ_{≥1}$ such that there exists a natural number $m$, such that for every $n > m$, there is a prime $p \in \mathbb{N}$ satisfying $cn > p > n$.

This is the "usual" way of defining $A$.

However, I would actually go one step further.

Consider that most people avoid lambda abstraction in function definitions. So rather than

Let $f = \lambda x\in \mathbb{R} : x^2.$

we would probably say the following:

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be given as follows.

$$f(x) = x^2$$

Sure, its a bit longer, but it is also a lot easier to understand.

For the same reasons, I think set-builder notation should be avoided in most circumstances, especially definitions, because it is basically lambda abstraction in disguise. For example, $\{x \in X \mid \varphi\}$ is analogous to $\lambda x \in X,\varphi$.

Therefore, I would probably define $A$ as follows.

Let $A \subseteq \mathbb{R}_{\geq 1}$ denote the unique set such that the following are equivalent.

• $c \in A$
• there exists a natural number $m$, such that for every $n > m$, there is a prime $p \in \mathbb{N}$ satisfying $cn > p > n$.

This has the advantage that its clear how to actually use the definition. Suppose we know that $\varepsilon \in A$, where $\varepsilon$ is some potentially complicated expression of type $\mathbb{R}_{\geq 1}$. Then we may conclude that that big long condition holds, except with the expression $\varepsilon$ substituted for $c$. Conversely, suppose we know that that big long condition holds, for some expression $\varepsilon$ is substituted for $c$. Then we may conclude that $\varepsilon \in A$.