There are a couple of notation-related matters that I find myself clumsily working around quite often when writing proofs, and I was hoping for some help with dealing with them:
Let's say that $L$, $L_1$ and $L_2$ are regular languages.
I claim:
$w \in (L_1 \cup L_2) \cdot L \iff \exists w_1 \in (L_1 \cup L_2), w_2 \in L$ such that $w=w_1\cdot w_2$.
So I have two questions about this:
- How should I handle that "there exist several $w_i$ such that…"? That is to say: is it okay to separate them with a comma as I did ($\exists w_1 \in …, w_2 \in …$), or would it be better to add a new $\exists$ per $w_i$ (e.g. $\exists w_1 \in … \wedge \exists w_2 \in …$)?
- Is there any kind of standard notation to use as a substitute for "such that"? I gather that $|$ and $:$ are used in set-builder notation and not in cases such as this one; so it'd be off to write something like: "$w \in (L_1 \cup L_2) \cdot L \iff \exists w_1 \in (L_1 \cup L_2), w_2 \in L: w=w_1\cdot w_2$". Is that correct?
Looking at some questions & answers on SE and on Google, my understanding was that the common consensus seems to be that writing "such that" is the preferred method, usually, but my native language is written RTL, and that makes it a bit awkward sometimes when I want to add "such that" in the middle of a LTR mathematical statement (I usually find myself adding line breaks in unfortunate places to deal with it), so I was hoping for some clear, creative solution based on standard notation for this. - Does it look okay if I write something like: "$\exists w_1, w_2$ such that $w_1\in … \wedge w_2 \in … \wedge w = w_1 \cdot w_2$"? That is, to concentrate all the "exists" in one place, and all their "descriptions" in another?
I get a strong sense that I'm overthinking it all, but I dislike the way a lot of my notation usages feel so clumsy to me. If you can see some sort of root cause (maybe a mindset issue or something more general like that) for my confusion & are able to point it out, that'd be great, too.
Best Answer
I’d write your example as
$$w \in (L_1\cup L_2)\cdot L \iff \exists w_1 \in (L_1\cup L_2): \exists w_2 \in L: w_1\cdot w_2 = w$$
The “such that “ is implicit in the final “:” (after the quqtifiers, before the term) for me, it need not be written out in natural language, and I prefer to give each variable it’s own quantifier, though other texts will do it differently. It depends also on how strictly “first order logic” you want to be, some will even not like the bounded quantifers and replace a statement like $\exists x \in A:P(x)$ by $\exists x: (x \in A) \land P(x)$ as well.
Like I said, it’s a matter of taste, largely. Just make sure it’s understandable.