Quantifiers

In predicates, we often don't want to make a statement about just one specific value, but about all of them or at least one. For this, we use quantifiers.

Universal Quantifier (∀)

Meaning: "P(x) is true for all x". That is, the predicate is true for every possible value.

Example: ∀x: "x^2 ≥ 0". The square of every number is non-negative, so this is a universal statement.

Existential Quantifier (∃)

Meaning: "there exists at least one x for which P(x) is true". That is, there is a value where the predicate is true.

Example: ∃x: "x > 10". This is true, because for example x = 11 works.

Combining Quantifiers

This means: for every x there exists a y that is greater than it. This is true because for any number we can find a larger one.

Negation and Quantifiers

• ¬(∀x P(x)) ≡ ∃x ¬P(x) → if not everything is true, then there is a counterexample.
• ¬(∃x P(x)) ≡ ∀x ¬P(x) → if there is no such, then it's false everywhere.

Summary

With quantifiers, we can formulate general and existential statements. The two basic quantifiers: ∀ (all) and ∃ (exists).

Practice Exercise