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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.

Asaf Karagila
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rdavid
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  • Not clear what you mean. Is $R$ meant to stand for $\mathbb R$, the set of real numbers? Something else? And, if it is, what does it mean to say that "for any real number $r$ we have $\sum 2r=2\sum r$."? What does $\sum r$ mean? – lulu Nov 19 '21 at 22:13
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    The $\forall$ symbol is called the universal quantifier and "binds" the variable $r$ in the formula that you call the second part. Without the first part the second part would be said to have $r$ as a free variable. Combined, the meaning of $r$ is bound to mean all $r\in\mathbb R$. – John Douma Nov 19 '21 at 22:14
  • Please use MathJax for the formulas in your MSE questions. – Rob Arthan Nov 19 '21 at 22:15
  • It'd be helpful to know more about what you're writing. – Karl Nov 19 '21 at 22:27
  • So, John Douma, any time i use a phrase that uses the universal quantifier, such as "for all x in set A", maybe i might call that a "binding expression"? Thoughts? – rdavid Nov 19 '21 at 22:28
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    @rdavid It can be referred to as a variable binding. – John Douma Nov 19 '21 at 23:18
  • Thank you, @JohnDouma. – rdavid Nov 20 '21 at 03:34
  • I’ve seen the first part called the prefix, and the second part called the matrix. This usually comes up when defining prenex normal form (which you can look up in Wikipedia or elsewhere). – Mitchell Spector Nov 20 '21 at 10:24
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    Thank you for introducing me to a new concept, @MitchellSpector. I looked up the Wikipedia article on PNF. Great learning. – rdavid Nov 22 '21 at 14:46

1 Answers1

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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".

aschepler
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  • I like that. So maybe i could call it a "quantifying expression"? – rdavid Nov 19 '21 at 22:49
  • @rdavid I think it's more natural/common to call that entire expression the "quantifier", for example "$\forall r\in D:P(r)$" or "$\forall r\in D(P(r))$" depending on how you like your notation may be said to be a quantifier, "$\forall r\in D$" followed by the predicate "$P(r)$". Predicate meaning just a statement, something that can either be true or false, like $r=3$. It can be informally considered a function which is true or false depending on $r$. I've heard the symbol $\forall$, $\forall r$, and $\forall r\in R$ all be referred to the quantifier part of the expression, as opposed to... – Pineapple Fish Nov 20 '21 at 03:50
  • @rdavid (con) the predicate part, so I guess it's a bit ambiguous. see https://math.stackexchange.com/questions/1024280/most-ambiguous-and-inconsistent-phrases-and-notations-in-maths?noredirect=1&lq=1 . In the end you may want to write out explicitly which of the above expressions applies if risks not being clear. Like aschepler said, that part of the expression is not actually unique, since something like "there exists x in D such that P(x)" can always be rewritten as "there exists x (emphasis on the fact that quantification is no longer restricted to the set D) such that x is in D and P(x)." – Pineapple Fish Nov 20 '21 at 03:58
  • It works because changing the condition from $P(x)$ to "$P(x)\text{ and }x\in D$" means that you can quantify over everything, except if $x\not\in D$ then it will turn the statement '$P(x)\text{ and }x\in D$' false and that value of x which is not in $D$ may be immediately discarded, if $x\in D$, then, well, it boils down to the usual statement. Similarly for $\forall$, but you must use '$x\not\in D\text{ or }P(x)$', aka $x\in D\to P(x)$ instead of the '$P(x)\text{ and }x\in D$' that you use for $\exists$. see https://en.wikipedia.org/wiki/Bounded_quantifier – Pineapple Fish Nov 20 '21 at 04:15
  • Hope that explanation helps, I don't know what your background knowledge is. Anyways, both just 'quantifier' (risks ambiguity) and 'variable binding' (risks referring to a process) or probably 'binding expression' (risks sounding too close to expression) are all actually fine, as long as you provide context. They all have some drawbacks so just use whatever feels most comfortable for you. Feel free to ask anyone for more clarification. – Pineapple Fish Nov 20 '21 at 04:24
  • Thank you, @PineappleFish. Very helpful. – rdavid Nov 22 '21 at 14:52