Terminology
Predicate Logic
The public will believe anything, so long as it is not founded
on truth. (Edith Sitwell)
Terminology . . . . . . . . . . . . .
Quantifiers . . . . . . . . . . . . . .
Negating Quantifiers . . . . . . .
Definition of Implies . . . . . . . .
Nested Quantifiers . . . . . . . . .
Examples of Nested Quantifiers
Complex Nested Quantifiers . .
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A predicate is a statement with variables, e.g., Boolean expressions,
conditions, tests.
With values to variables, a predicate is T or F.
P (x) denotes a predicate with one variable.
Q(x, y) denotes a predicate with two variables.
Examples: x > 3, x = y + 3, x y = z
We might define P (x) to be x > 3,
Q(x, y) to be x = y + 3, and
R(x, y, z) to be x y = z.
The domain specifies all possible values of the variables. Unless otherwise
stated, the domain is R, the real numbers.
CS 2233 Discrete Mathematical Structures
Predicate Logic – 2
Quantifiers
We form propositions from predicates by assigning values to the variables or
by quantification.
Universal quantification:
∀x P (x) is T if P (x) is T for all values of x.
–
Examples: ∀x (x > 3), ∀x (x2 ≥ 0)
Existential quantification:
∃x P (x) is T if P (x) is T for some value of x.
–
Examples: ∃x (x > 3), ∃x (x2 < 0)
CS 2233 Discrete Mathematical Structures
1
Predicate Logic – 3
2
Negating Quantifiers
Nested Quantifiers
¬∃x P (x) is equivalent to ∀x ¬P (x)
Ex: ¬∃x(x2 < 0) is equiv. to ∀x(x2 ≥ 0)
Note: the negation of x2 < 0 is x2 ≥ 0.
– Ex: ¬∃x(x = x2 ) is equiv. to ∀x(x 6= x2 )
Note: the negation of x = x2 is x 6= x2.
–
¬∀x P (x) is equivalent to ∃x ¬P (x)
–
–
Ex: ¬∀x(x = x2) is equiv. to ∃x(x 6= x2 )
Ex: ¬∀x(x < x + 1) is equiv. to ∃x(x ≥ x + 1)
Note: ¬(x < x + 1) is x ≥ x + 1.
CS 2233 Discrete Mathematical Structures
Predicate Logic – 4
Definition of Implies
CS 2233 Discrete Mathematical Structures
Predicate Logic – 6
Define P (x) implies Q(x) as ∀x(P (x) → Q(x)).
Note: ∀x(P (x) → Q(x)) is equivalent to ¬∃x(P (x) ∧ ¬Q(x)).
Examples of Nested Quantifiers
P (x) implies Q(x) is true if, whenever P (x) is true, Q(x) is also true.
P (x) implies Q(x) is false if there is any counterexample x = a where P (a) is
true and Q(a) is false.
The sum of two positive numbers is positive.
–
Example 1: x < 0 implies x − 1 < 0.
Every number has an additive inverse.
–
Example 2: x < 0 does not imply x + 1 < 0.
CS 2233 Discrete Mathematical Structures
Predicate Logic – 5
∀x∃y(x + y = 0)
There is a number such that this number times any other number results in
zero.
–
∀x∀y((x > 0 ∧ y > 0) → (x + y > 0))
∃y∀x(x y = 0)
There exists two negative numbers that add up to −3.
–
∃x∃y(x < 0 ∧ y < 0 ∧ x + y = −3)
CS 2233 Discrete Mathematical Structures
3
Predicate Logic – 7
4
Complex Nested Quantifiers
The definition of limits: lim f (x) = b if
x→a
∀ǫ ∃δ ∀x (0 < |x − a| < δ → |f (x) − b| < ǫ)
where ǫ and δ are positive reals and x is real.
The definition of big-Oh (later in course): f (n) is O(g(n)) if
∃k∃c∀n(n > k → f (n) < c g(n))
where the domain is R, the real numbers.
CS 2233 Discrete Mathematical Structures
Predicate Logic – 8
5