A sentence like "$x^{2} > 4$" is not a statement until we know what $x$ is. Quantifiers turn such sentences into statements by saying how many values of $x$ we mean. They are the language of every theorem you will prove, and they connect to the equivalences you just learned.
Key idea
A predicate $P(x)$ is a sentence about $x$ that becomes true or false once $x$ is chosen from a domain. The universal statement $\forall x\, P(x)$ ("for all $x$") is true when $P(x)$ holds for every $x$ in the domain. The existential statement $\exists x\, P(x)$ ("there exists $x$") is true when $P(x)$ holds for at least one $x$.
Why it works
On a finite domain, the quantifiers are just long "and"s and "or"s. If the domain is $\{1, 2, 3\}$, then $\forall x\, P(x)$ means $P(1) \wedge P(2) \wedge P(3)$, and $\exists x\, P(x)$ means $P(1) \vee P(2) \vee P(3)$. That explains how to check them: one failure sinks a "for all," and one success is enough for a "there exists." It also explains why the domain matters: change the list of things being joined, and the truth can change.
| Statement | positive integers | integers | rational numbers | real numbers |
|---|---|---|---|---|
| $\exists x\, (x^{2} = 2)$ | F | F | F | T |
| $\exists x\, (2x = 1)$ | F | F | T | T |
| $\forall x\, (x^{2} \ge x)$ | T | T | F | F |
| $\exists x\, (x + 5 = 2)$ | F | T | T | T |
Worked examples
Worked example: Checking quantified statements on a finite domain
Let the domain be $D = \{1, 2, 3, 4\}$. Decide $\forall x\, (x^{2} > x)$ and $\exists x\, (x^{2} = 2x)$.
- For $\forall x\, (x^{2} > x)$, look for a failure. At $x = 1$: $1^{2} = 1$, which is not greater than $1$. One failure is enough: false.
- For $\exists x\, (x^{2} = 2x)$, look for a success. At $x = 2$: $4 = 4$. True.
- Notice the asymmetry: the first needed one counterexample, the second one example.
Answer: false, true
Worked example: Writing English with quantifiers
The domain is the students in a class, and $S(x)$ means "$x$ passed the test." Write "Every student passed," "Some student passed" and "No student passed."
- "Every" is $\forall$: $\forall x\, S(x)$.
- "Some" means at least one: $\exists x\, S(x)$.
- "No student passed" means every student did not pass: $\forall x\, \neg S(x)$. (It is also $\neg \exists x\, S(x)$, as the next lesson shows.)
Answer: $\forall x\, S(x)$, $\exists x\, S(x)$, $\forall x\, \neg S(x)$
Common mistake
Ignoring the domain. "$\exists x\, (x^{2} = 2)$" is true over the real numbers ($x = \sqrt{2}$) but false over the rational numbers, because $\sqrt{2}$ is irrational. Check: before deciding, ask which numbers $x$ may be, and whether your example or counterexample belongs to that domain.
In mathematical writing the quantifier is often hidden. "If $n$ is even, then $n^{2}$ is even" really means "for every integer $n$, if $n$ is even, then $n^{2}$ is even."