Quantifiers – Structure, Universe, and Scope

 

If you’re thinking about learning math beyond the standard high school curricula, chances are you’ve either seen quantifiers already or will see run into them very soon. Getting comfortable with quantifiers is critical for studying higher levels of mathematics, but fortunately they are quite simple. Quantifiers appear before a statement, and tell you what that statement applies to. As an example, we’ll consider the statement “\(x\) is an even number,” and write it as \(E(x)\) for shorthand. (E.g. \(E(2)\) stands for “\(2\) is an even number” and \(E(5)\) stands for “\(5\) is an even number.”)

 

There are two types of quantifiers. The first type is the universal quantifier, written as \(\forall\), and followed by a variable which will give us a name to reference. The universal quantifier is usually spoken in English as “for all,” and as this might suggest it tells you to apply the statement to “everything.” (We’ll discuss exactly what “everything” means a little later.) As an example, we might say \(\forall x\ E(x)\). In other words, this says “For all \(x\), \(x\) is an even number.”

 

The second type is the existential quantifier, written as \(\exists\). In English we would say “there exists.” It uses the same structure as a universal quantifier, so we could say \(\exists x\ E(x)\). This would say “There exists an \(x\) that is an even number.”

 

It is important to observe that the universal quantifier and the existential quantifier are in some sense opposites of each other: If \(P(x)\) is any statement about \(x\), then \(\forall x\ P(x)\) is true, i.e. every \(x\) satisfies \(P(x)\), if and only if the statement “There exists an \(x\) such that \(P(x)\) is not true” is true. Similarly, \(\exists x\ P(x)\) is true if and only if “For all \(x\), \(P(x)\) is not true” is false. (Often the symbol \(\neg\) is used to denote the negation of something, e.g. \(\neg P(x)\) would be read as “\(P(x)\) does not hold.” For example, we could rewrite “There exists an \(x\) such that \(P(x)\) is not true” as \(\exists x\ \neg P(x)\).)

 

The Universe

    When reading quantifiers, it is absolutely critical that you know what the “universe” is in context. Here the universe is some collection of objects, not necessarily all of them, and we only care about the statement applied to that specific collection. For example, consider the following statement:

\[\forall x\ \forall y\ \text{If }x<y\text{ then }\exists z\ \text{with }x<z\text{ and }z<y\]

 

In English, this says “For every \(x\) and \(y\), if \(x\) is less than \(y\), then there is a \(z\) with \(z\) greater than \(x\) and less than \(y\).” Is this statement true?

 

The answer is actually a little complicated. Suppose our universe is all of the integers, or whole numbers. (Think \(1\), \(-5\), \(372\), and so on, but no simplified fractions with a denominator that isn’t \(1\) are allowed.) Then this statement is false: If \(x=0\) and  \(y=1\), then \(x<y\) is certainly true, but there is no whole number \(z\) which is bigger than \(0\) but smaller than \(1\). Notice, though, that \(\frac{1}{2}\) does satisfy these inequalities, but it is not in our universe. Now if we let our universe be all possible fractions, not just the integers, then the statement becomes true. For any fractions \(x\) and \(y\) with \(x<y\), consider \(\frac{x+y}{2}\). This is clearly a fraction, as it is the sum of two fractions divided by \(2\), so it is in our universe. Furthermore, \(x<\frac{x+y}{2}\) and \(\frac{x+y}{2}\), so it satisfies our inequalities. (To see this last fact, note that \(2x=x+x<x+y\) since \(x<y\) and \(2y=y+y>x+y\) for the same reason.)

 

Keep this in mind when dealing with quantifiers: a statement can easily be true in one universe and false in another. In fact, some areas of mathematics have healthy research communities dedicated to finding exactly which “universes” a specific statement holds for. (The classification of all finite simple groups is a famous example, if a bit complicated: https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups )

 

The Scope of a Quantifier

 

One final thing to keep in mind when reading quantifiers: the scope. Consider the following statement:

 

\[\forall x\ \text{if }\exists z\ z<x\text{ then }\forall y\ y<x+y\]

 

The scope of a quantifier is the amount of the statement that the quantifier applies to. In the above example, \(\forall x\) applies to the whole statement, \(\exists z\) only applies to the portion between if and then, and the \(\forall y\) only applies to the part after then. In general it’s good practice to use parentheses/brackets/braces to separate the scope of a variable for absolute clarity. We could rewrite the above as:

 

\[(\forall x\ \text{if }\{\exists z\ z<x\}\text{ then }[\forall y\ y<x+y])\]

 

The meaning of this statement has no longer changed, but it is now much more clear at a quick glance which quantifiers apply where.

 

Different scopes, like different universes, can change the meaning of a statement. When writing down statements it’s best to be as clear as possible with your quantifiers to avoid confusion.