What do you call the set definition part of a formula?

Suppose I have a formula with two parts: First is a description of the element and set on which an equation will depend, and the second is the equation itself.

Example:

First, the formula quantifies the element and set on which the equation depends: $$∀r∈R$$

Second, the formula shows the equation itself:

$$∑2r = 2∑r$$

Full formula:

$$∀r∈R, ∑2r = 2∑r$$

Is there a specific word/name for what we call that first part of the formula? Is it just the 'set definition' portion of the formula? I am a tech writer and it would be useful for me to know if there is an actual technical word for this or if i just have to resign to circumlocution.

Note: Please don't let my example formula (shown above) distract from my question. The actual formula itself doesn't matter. It could be any equation with any sets/elements. I'm just looking for words to describe what i'm writing.

Solution 1:

I believe you're looking for the term "quantifier". A first order logic system has formulas involving the universal quantifier $\forall$ "for all" and the existential quantifier $\exists$ "there exists", to relate a variable to a predicate formula.

A strict logic grammar will usually only use these to introduce the variable, and not associate a set also. But in math writing less focused on the logic structure itself, yes, we often see

$$ \forall v \in S: P(v) $$

meaning

$$ \forall v: v \in S \rightarrow P(v) $$

and

$$ \exists v \in S: Q(v) $$

meaning

$$ \exists v: v \in S \land Q(v) $$

Or similarly with $\subseteq$ in place of $\in$, and perhaps other simple restrictions. I think it would still be fine to call the initial part the "quantifier".