# What character can replace word “let” in proofs?

For example, suppose I have a line of a proof introducing new “variable” $x$: $$\textrm{Let}\:\:x\in f(y)$$ I am looking for ways to express the word “let” in this context and I would like to avoid using natural language because Math is itself a universal language for expressing complex ideas.

I used to use a character like a right square bracket or similar to it. I once encountered it somewhere, but I don't know if it is a common practice. For example, I would write that line like this: $$\sqsupset x\in f(y)$$

Now I failed to find this character anywhere, neither on the Web nor in Unicode symbol set. Instead, I discovered some similar symbols like $\buildrel \text{def}\over=$, or $:=$, or $\buildrel\triangle\over=$, or $:\Leftrightarrow$ in Wikipedia, but those are very limited and not so much useful in my case.

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Personally, I find that replacing everything by symbols does not improve the readability, so I would advice against using a symbol for "let". – Tobias Kildetoft Jul 26 '13 at 21:12
There is an opposite view that words are often better than symbols in aspects of proof writing. This is especially so if the symbols are not widely used and common. Of course symbols are often necessary. So one can write: "Let $x$ belong to the set $f(y)$." – Ronnie Brown Jul 26 '13 at 21:14
I think that it's ugly to begin a sentence with a symbol. – user63181 Jul 26 '13 at 21:25
With all due respect, the OP is looking for a standard symbol in this context (not opinions about whether this is a good practice or not). Perhaps proof theory people can help. – Lord Soth Jul 26 '13 at 21:31
Several standard settings for formal proofs (in the sense of proof theory) do not have a way of formally writing "Let $x$ ...", so @LordSoth's suggestion does not quite help here. (Instead, we have rule for addition or removal of quantifiers, but this is a different issue.) Plus, the question is not really asking for answers in that setting, given the example in the last paragraph. (OT: I also strongly advice against the practice of excessive symbolism instead of words. It tends to decrease readability, and does not make the text look more "formal" or "mathematical".) – Andrés E. Caicedo Jul 26 '13 at 21:39

I would like to avoid using natural language because Math is itself a universal language for expressing complex ideas.

Honestly, I think this is a bad (by which I mean non-mathematical) reason to do anything. You will have great difficulty introducing any symbol into any widespread use, and consequently you will not be able to use such a symbol in any piece of work you wish to be taken seriously (because nobody will be able to read it!). I also don't recommend you teach people bad habits.

On the other hand, if these are just for personal notes, by all means invent your own symbol. I use lots of imprecise squiggly arrows, equals signs with quotation marks round them, equals signs decorated with question marks, and the like. I often use := to conjure a symbol into existence at the same time as defining it (because, unlike when programming, I don't need to declare my variables and I don't often redefine them in the same 'subroutine'), or simply =. I've seen people write an equals sign with "def" or "$\triangle$" above it too.

This is not unlike how real mathematicians work with each other. When two people collaborate, it's very convenient for there to be an implicit assumption along the lines of "every time I say X, until we solve this problem, I mean this particular object", or vague terminology like "nice" to describe classes of objects that you can't quite pin down. But of course, once it comes to a seminar or a paper, you start from scratch, (mostly) give everything real, sensible words, and don't force your audience to learn a page of jargon and squiggles when there's perfectly good English available for it.

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This is so true. I once had a student who annoyed me to no end by insisting to write all the proofs in Mathemateze. Completely unreadable! – PA6OTA Jun 5 '14 at 16:46

If you're seeking to completely eliminate natural language from your proof, then it seems to me that this is synonymous with seeking to write a formal proof. So, let me assume that, in natural language, the statement you're trying to prove is something of the form, "Let $x$ satisfy the property $\mathbf\Phi(x)$. Then $\mathbf\Psi(x)$.". Here $\mathbf\Phi(x)$ and $\mathbf\Psi(x)$ represent natural language statements meant to be interpreted as first-order propositions in which $x$ (and possibly other variables) occurs free. I'll use $\mathbf\Phi$ and $\mathbf\Psi$ (bold-face) to represent the natural language statements, and $\Phi$ and $\Psi$ (light-face) to represent the corresponding formal statements. So, the first-order proposition you're trying to prove is $\forall x. \Phi(x) \rightarrow \Psi(x)$. Now, in the System LK, two of the rules you'll use for proving this proposition are $(\mathbf{\forall R})$, which here will take the form:

$$\genfrac{}{}{1pt}{}{\Gamma \vdash \Phi(y) \rightarrow \Psi(y),\Delta}{\Gamma \vdash \forall x.\Phi(x) \rightarrow \Psi(x), \Delta}$$

and $(\mathbf{\rightarrow\kern {-1ex} R})$, which here will take the form:

$$\genfrac{}{}{1pt}{}{\Gamma, \Phi(y) \vdash \Psi(y), \Delta}{\Gamma \vdash \Phi(y) \rightarrow \Psi(y), \Delta}$$

These two deduction steps are what take the place of saying "let".

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I've got a suggestion for you: Invent a completely new symbol which looks exactly like the word "Let". Then use that symbol instead of "Let". This way your desire to use a special symbol is fulfilled, while anyone seeing it will still immediately know what you mean.

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While I appreciate the humor (I chuckled), I'm not sure about this being an answer instead of a comment. – mixedmath Jul 31 '13 at 18:49
Symbols are language independent. – Kasper Jan 13 '14 at 1:13
@Kasper: Actually they aren't. Some languages use a decimal point, others use a decimal comma. In the USA the common number multiplication symbol is $\times$, in Germany it's $\cdot$ (due to foreign inferences, the $\times$ is also sometimes used in general writings, but never when learning mathematics). I'm not sure whether the colon is used for division in the USA (my impression is that it isn't used), but it is the standard division sign in Germany. I guess you'll find other differences as well. – celtschk Jan 25 '14 at 11:12

I agree with others who advise against using an obscure symbol rather than a simple English word.

However, you asked what symbol to use; you didn't ask whether using it is a good idea.

It seems to me that ...

"Let $x \in A$. Then $x$ has the property ..."

can be replaced by

"$\forall x \in A$, $x$ has the property ..."

So, maybe the symbol $\forall$ will serve your needs (in some situations, at least). It's not as obscure as the symbol you mentioned, but I still think the word "let" is better. Your decision, though.

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Depending on the context, it could also be $\exists{x}\in{A}$. – Ataraxia Jul 27 '13 at 2:29
In fact, it could be neither: "Let $x\in A$ be such that ..." does not mean that there is an $x$ such that ..., and does not mean that every $x$ ... – Andrés E. Caicedo Jul 27 '13 at 2:39
@Andres. True. But "Let $x \in A$ be such that ..." is not the phrase I suggested replacing. – bubba Jul 27 '13 at 2:54
@bubba True. The comment was really aimed at the OP, lest they misunderstand your suggestion. – Andrés E. Caicedo Jul 27 '13 at 2:59
I really like this answer. And "Let $x \in A$ be such that $\Phi(x)$ ... is equivalent to $\forall x \in A: \Phi(x) \Rightarrow ....$. – DaveBall aka user750378 Jul 31 '13 at 18:43

Note that both × and · are relatively language-independent notations for multiplication. One is "Cross Multiplication", the other is "Dot Multiplication"; both devolve to "simple linear multiplication" when being applied to degenerate (i.e. 1 by 1 (by 1 by ...)) matrices, and either will generally be recognized as such by any technical person in a locale that uses Arabic-derived mathematics. In other words, Western countries and Near-Eastern countries, and most of their former colonies and subject-states - which pretty much covers everything but China (I don't know enough about the impact of the "modernization/westernization/digitization" of communications on their technical notations to comment on them).

Both mean entirely different things when the context is multidimensional matrices, however. That is, in fact, WHY specific symbols are used. If you just say "Matrix A times Matrix B", it is not clear whether you mean cross-multiplication or dot-multiplication - or even something else entirely. Sure, there are conventions for ad-hoc interpretation; but for a precise and unambiguous statement, a narrowly-defined (and thus, of restricted use) symbolic language is required.

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Thank you! Following this thought it's obvious that the word let itself is very clear and unambiguous and there's no reason to use a specific symbol, it doesn't add anything. Nice addition! – eush77 Oct 25 '15 at 11:52