Universal and existential quantifiers and basic predicate logic notation used beyond propositional logic.
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- What does the universal quantifier symbol ∀ represent?
- 'for all' - asserts that a property holds for every element in the domain
- What does the existential quantifier symbol ∃ represent?
- 'there exists' - asserts that a property holds for at least one element in the domain
- What is a predicate in predicate logic?
- A function that assigns true or false to objects; typically written P(x), Q(x,y), etc.
- What does the notation P(x) mean?
- P is a predicate; x is a variable; truth value depends on what x represents
- What is the domain or universe of discourse in predicate logic?
- The set of all objects that variables can refer to
- What is the negation of the statement forall x, P(x)?
- ∃x ¬P(x)
- What is the negation of the statement there exists x, P(x)?
- ∀x ¬P(x)
- Translate to predicate logic: All humans are mortal.
- ∀x (Human(x) → Mortal(x))
- Translate to predicate logic: Some dogs are friendly.
- ∃x (Dog(x) ∧ Friendly(x))
- Translate to predicate logic: No integers are both odd and even.
- ∀x (Integer(x) → ¬(Odd(x) ∧ Even(x)))
- What is a free variable in a formula?
- A variable not bound by any quantifier; not in the scope of ∀ or ∃
- What is a bound variable in a formula?
- A variable in the scope of a quantifier (∀ or ∃)
- In the formula ∃x P(x, y), which variable is free and which is bound?
- x is bound; y is free
- What is an open formula in predicate logic?
- A formula with at least one free variable; truth value depends on what the free variables represent
- What is a closed formula or sentence in predicate logic?
- A formula with no free variables; every variable is bound by a quantifier
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