What is the Correct Math Notation for - "For all $i=1$ to $n$"

Question

I want to make sure I am writing correct math notation...

I want to write: "For all $i$ from 1 to $n$" (It's been a while...)

How do you write this?

I get this so far...$\forall_{i=1,\ \mbox{to}\ n}$ Is this right???

Answer 1

Properly speaking, "For all $i$ from $1$ to $n$" is perfectly legitimate mathematics, and is often preferable from the standpoint of exposition. For example, it might be preferable when writing a paper. The use of symbols like $\forall \exists! \ni \wedge\leadsto $ can quickly devolve into a jumbled mess and should be thought of more as "mathematical shorthand" than proper exposition.

If you insist on more compact notation, something like

$$\forall i \in \{1, \dots, n\} \text{ or } \forall_{i \in \{1, \dots, n\} } \text{ or } \forall 1 \leq i \leq n \text{ or } \forall i = 1..n$$

would be acceptable (I personally like the first two). There isn't really a summation notation version--something like $\forall_{i=1}^n $ looks weird to me.

I would also note that something like $\forall 1 \leq i \leq n$ can introduce some ambiguity....is $i$ supposed to be an integer or a real number? By choice of $i$ as the variable name, I would assume it is an integer, but I would be unable to know that just from the notation.