# Translate “any open interval” and “any closed interval” from English to math symbols.

first-order-logiclogicpredicate-logic

In my intro to real analysis book, I came up with the following lemma (which is easy to prove) to help with an exercise.

For any open interval $$K$$, if any closed interval $$S$$ that is a subset of $$K$$ has the property that for any $$x \in S$$:$$\varphi(x)$$, then for any $$x\in K$$, we must have $$\varphi(x) \quad(\dagger)$$

For the proof, consider any $$x \in K$$. Now, consider the set $$\{x\}$$. This set is a closed set. Therefore, by assumption $$\varphi(x)$$.

Question: Could someone help me translate the English of $$(\dagger)$$ into the appropriate math notation?

How does one encode "$$K$$ is any open interval" and "$$S$$ is any closed interval"? Are these strictly topological notions that require new notation? Or is there a clever way that uses basic quantifiers and inequalities?

For example, does the following syntax work?

$$\forall K \Bigg[\bigg(\Big[\exists a,b \in \mathbb R: \forall x (a \lt x \lt b \rightarrow x \in K) \Big] \text { and } \Big[\forall S\color{blue}{\big(}\color{red}{(}\exists a,b \in K: \forall x (a \leq x \leq b \rightarrow x \in S)\color{red}{)}\rightarrow \forall x \in S: \varphi (x)\color{blue}{\big)}\Big]\bigg) \rightarrow \forall x \in K: \varphi (x) \Bigg]$$

Edit:

I think it may be necessary to add a further specification to each of the conjuncts in the overarching antecedent.

For example, in the statement $$\exists a,b \in \mathbb R: \forall x (a \lt x \lt b \rightarrow x \in K)$$, I have not ensured that $$K=\{x \in \mathbb R : a \lt x \lt b\}$$. Rather, I have only ensured that $$\{x \in \mathbb R : a \lt x \lt b\} \subseteq K$$. To guarantee equality, I would have to add the condition that $$\forall x ( x \leq a \text{ or } x \geq b \rightarrow x \notin K)$$.

A similar extra condition would have to be stipulated for $$S$$.

I want to make a few notes:

• Bear in mind that stuff like $$K, S$$ are just names. You don't need to give coordinates/bounds for your intervals necessarily, unless you want to specifically reference those bounds. It's the same with any set: unless you need to reference particular features or elements, don't overcomplicate details.

• You can assume, as needed, any definitions in statements like these. Your translation need not somehow include a definition of what it means to be an open interval. If need be, you can literally just say what $$K,S,$$ etc. are, if doing so is cumbersome symbolically. There generally isn't issue with this. Mathematical language is in part about communication, after all.

• (After all, going to an extreme, we shouldn't have to define basic notions like set inclusion, set membership, or even basic notations like "what is a real number" for stuff like this to be meaningful. Those are handled elsewhere; make sufficient assumptions on what the reader knows so that we can concisely yet precisely communicate what we need to.)

So, to translate

For any open interval $$K$$, if any closed interval $$S$$ that is a subset of $$K$$ has the property that for any $$x \in S$$:$$\varphi(x)$$, then for any $$x\in K$$, we must have $$\varphi(x) \quad(\dagger)$$

I would simply go with

\begin{align*} &(\forall K \subseteq \Bbb R \text{ an open interval})(\forall S \subseteq K \text{ a closed interval})(\forall x \in S)(\varphi(x)) \\ &\implies (\forall x \in K)(\varphi(x)) \end{align*}

Broken down a bit:

• Let $$K \subseteq \Bbb R$$ be some open interval...
• ...where for any subset $$S \subseteq K$$ where $$S$$ is a closed interval...
• ...and each $$x \in S$$ has property $$\varphi(x)$$.
• Then we say that property $$\varphi(x)$$ holds for all $$x \in K$$.

If you want to avoid the writing things as text, you could then introduce notation representing $$K,S$$ as intervals formally: $$K = (k,\ell)$$ and $$S = [s,t]$$ below:

$$(\forall (k,\ell) \subseteq \Bbb R)(\forall [s,t] \subseteq (k,\ell) )(\forall x \in [s,t])(\varphi(x)) \implies (\forall x \in (k,\ell))(\varphi(x))$$

(Or something to this effect anyhow, it's hard for me to parse through what you wish to translate, but I think I'm somewhere close at least.)