Quantification is some jargon from formal logic. There is the universal quantifier written as ∀ in a context such as “∀x(x=x)” which is taken to mean that for all x, x is x. Tarski wrote “∧x(x=x)” in the same context. The other quantifier is ∃ where “∃x(x>4)” means that for some x, x is greater than 4. Tarski wrote “∨x(x>4)”.
To quantify over sets means that the variable following the quantifier is to range over sets.
Unicode together with font mongers do not support Tarski’s notation well.
∀x∃y y mod x = 0.
∃y∀x y mod x = 0.
The first is true and the second is false.
John Maynard Keynes famously said: “In the long run we are all dead.”. Let’s pull that apart:
“∀x” is the universal quantifier and “∃y” is the existential quantifier.
If φ is some expression, perhaps involving y then.
∃yφ = ¬∀y¬φ
This is often taken as a definition for ∃ while ∀ is primitive.
Universal quantifiers commute with each other and so do existential quantifiers. They do not commute with each other.
“For every x (x divides y) for some y.” can be read as either:
“For every x ((x divides y) for some y).” or
“(For every x (blah blah x blah y)) for some y.”.
The meanings, as above, are different. Professional mathematicians often make this mistake.
∀ε (ε>0 → ∀x∃δ (δ > 0 ∧ ∀z(|x − z| < δ → |f(x) − f(z)| < ε))) ↔︎ The function f is continuous.
∀ε (ε>0 → ∃δ∀x (δ > 0 ∧ ∀z(|x − z| < δ → |f(x) − f(z)| < ε))) ↔︎ The function f is uniformly continuous.
One must be careful with quantifiers.