Quantifiers Flashcards
a declarative statement with variables, whose truth can be found by assigning “values” to the variables.
Predicate
The statement “x is taller than y.” is an example of
a predicate on x and y
The statement “x is divisible by 2.” is an example of
a predicate on x
Denoted using function notation, usually with capital letters
Predicates
Used to join propositions and predicates to form more complex statements
connectives
Used to denote that a predicate is always true.
Universal Quantifier (∀)
Used to denote that a predicate is sometimes true.
Existential Quantifier (∃)
¬∀x(P(x)) ≡
∃x(¬P(x))
¬∃x(P(x)) ≡
∀x(¬P(x))
“Let
P(x) ≡ “x is hot.”
Q(x) ≡ “x is cold.”
R(x) ≡ “x is expensive.”
Translate the following statement into quantifiers.
Something is hot.”
∃x(P(x))
“Let
P(x) ≡ “x is hot.”
Q(x) ≡ “x is cold.”
R(x) ≡ “x is expensive.”
Translate the following statement into quantifiers.
All things are expensive.”
∀x(R(x))
“Let
P(x) ≡ “x is hot.”
Q(x) ≡ “x is cold.”
R(x) ≡ “x is expensive.”
Translate the following statement into quantifiers.
Nothing is hot and cold.”
¬∃x(P(x) ∧ Q(x))
“Let
P(x) ≡ “x is hot.”
Q(x) ≡ “x is cold.”
R(x) ≡ “x is expensive.”
Translate the following statement into quantifiers.
Hot things are not cold.”
∀x(P(x) → ¬Q(x))
“Let
P(x) ≡ “x is hot.”
Q(x) ≡ “x is cold.”
R(x) ≡ “x is expensive.”
Translate the following statement into quantifiers.
If coffee is hot, then it is expensive.”
P(coffee) → R(coffee)
What is the justification for this statement?
Universal Instantiation on 1.
What is the justification for this statement?
Existential instantiation on 1.
What is the justification for this statement?
Universal Generalization on 1.
What is the justification for this statement?
Existential Generalization on 1.
