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
We have reviewed and checked the materials, but errors may still occur. The content is provided for educational purposes only, so use it at your own responsibility and verify with other sources if needed.
✨ Ask Lara — your AI study partner
Unlock personalized learning support. Lara can explain lessons, summarize topics, and answer your study questions — available from the Go plan and above.
Lara helps you learn faster — exclusive to ReadyTools Go, Plus, and Max members.
