Quantifiers

Beyond "And" and "Or", mathematical statements frequently use two special phrases called quantifiers: "there exists" and "for every" (also written "for all").

• "There exists" asserts that at least one member of a set has a stated property. For example, p: "There exists a rectangle whose all sides are equal" claims that at least one such rectangle can be found (a square, in fact) — it does not claim that every rectangle has equal sides.
• "For every" / "For all" asserts that every single member of a set has the stated property, with no exceptions. For example, p: "For every prime number p, $\sqrt{p}$ is an irrational number" claims this holds for every prime, without exception.

Quantifiers are written compactly using the symbols $\forall$ ("for all") and $\exists$ ("there exists").

The order of quantifiers changes the meaning — and can change the truth value. Compare:

1. For every positive number x, there exists a positive number y such that y < x. (True — however small x is, some smaller positive y can always be found.)
2. There exists a positive number y such that, for every positive number x, y < x. (False — this would require one fixed y smaller than every positive number, which is impossible.) …