# Theorem formulation “Given …, then …” or “For all …, …”?

When formulating a theorem, which of the following forms would be preferred, and why? Or is there another even better formulation? Are there reasons for or against mixing them in one paper?

Formulation 0: If $x\in X$, then (expression involving $x$).

Formulation 1: Given $x$ in $X$, then (expression involving $x$).

Formulation 2: For all $x$ in $X$ it holds that (expression involving $x$).

-
I used the 'proof-writing' tag because no other tag I found seemed appropriate, and I cannot create new ones. –  equaeghe Mar 21 '12 at 10:46
They're all equivalent. Use what reads best. If you can give a concrete example, perhaps we can help you choose. –  lhf Mar 21 '12 at 10:56
@lhf: I've added an extra subquestion regarding mixing them in one paper. –  equaeghe Mar 21 '12 at 11:22
Why not examine a few papers you like, and see what is done there? Then see if you can tell why you like them. –  GEdgar Mar 21 '12 at 13:11
"For all $x$ in $X$ it holds that $\dots$" sounds to me like something transliterated from a language other than English. –  André Nicolas Mar 21 '12 at 16:12

I guess I'm a simpleton, I always preferred a theorem stated memorably:

Theorem 1. Even numbers are interesting.

Compared to...

Theorem 2. If $x$ is an even number, then $x$ is interesting.

...which is long-winded; or...

Theorem 3. Given an even number $x$, $x$ is interesting.

Theorem 3 also has odd consonance (writing "...$x$, $x$ is ..." seems like bad style).

Theorem 4. For all even numbers $x$, we have $x$ be interesting.

This also suffers from peculiar consonance ("we have $x$ be interesting" is quite alien, despite being proper grammar).

Theorem 1 has a quick and succinct enunciation ("Even numbers are interesting"), the proof can begin with a specification "Let $x$ be an even number. Then [proof omitted]."

Also we see theorem 1 has additional merit: it avoids needless symbols.

There is one warning I should give: each theorem is different. Some theorems can be stated beautifully without symbols (e.g., theorem 1). Others cannot be coherently stated without symbols. There is not "iron law" on how theorems should be formulated; it's a case-by-case problem.

-

Formulation 0 confounds the binding of the symbol $x$ as a variable to $X$ with set inclusion $x\in X$.

The other two formulations are equivalent, but the second seems stronger and is more verbose, so for now I prefer Formulation 2.

-
Whu? 0 and 2 are obviously equivalent. Compare the following two definitions of continuity at $x_0 \in \mathbb R$: $$\forall \epsilon > 0 \exists \delta > 0 : |x-x_0| < \delta \implies |f(x)-f(x_0)| < \epsilon,$$ and $$\forall \epsilon > 0 \exists \delta > 0 \forall x\in (x_0-\delta,x_0+\delta): |f(x)-f(x_0)| < \epsilon.$$ –  kahen Apr 21 '12 at 11:40
@kahen: the ‘$\in$’ in ‘$\forall x\in(x_0-\delta,x_0+\delta)$’ is not set inclusion from a formal point of view; this is what I was referring to. (Cf. the often-made distinction between equality ‘$=$’ and definition ‘$:=$’.) –  equaeghe Apr 22 '12 at 16:15