Discrete Mathematics 1.4 Predicates and Quantifiers

The existential quantification of P(x) is the proposition "There is an x such that P(x)," "There is at least one x such that P(x)," or "For some xP(x)."

"There exists an element x in the domain such that P(x)." We use the notation ∃xP(x) for the existential quantification of P(x). Here ∃ is called the existential quantifier.

¬∃xQ(x) ≡ ∀x ¬Q(x).

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Predicates

Statements involving variables, such as "x > 3," "x = y + 3," "x + y = z," and "computer x is under attack by an intruder," and "computer x is functioning properly,"

The universal quantification of P(x) is the statement "P(x) for all values of x in the domain."

The notation ∀xP(x) denotes the universal quantification of P(x). Here ∀ is called the universal quantifier.We read ∀xP(x) as "for all xP(x)" or "for every xP(x)." An element for which P(x) is false is called a counterexample of ∀xP(x).

uniqueness quantifier, denoted by ∃! or ∃1.

The notation ∃!xP(x) [or ∃1xP(x)] states "There exists a unique x such that P(x) is true." (Other phrases for uniqueness quantification include "there is exactly one" and "there is one and only one.")

Precedence of Quantifiers

The quantifiers ∀ and ∃ have higher precedence than all logical operators from propositional calculus. For example, ∀xP(x) ∨ Q(x) is the disjunction of ∀xP(x) and Q(x). In other words, it means (∀xP(x)) ∨ Q(x) rather than ∀x(P(x) ∨ Q(x))

What do the statements ∀x < 0 (x2 > 0), ∀y != 0 (y3 != 0), and ∃z > 0 (z2 = 2) mean, where the domain in each case consists of the real numbers?

The statement ∀x < 0 (x2 > 0) states that for every real number x with x < 0, x2 > 0. That is, it states "The square of a negative real number is positive." This statement is the same as ∀x(x < 0 → x2 > 0). The statement ∀y != 0 (y3 != 0) states that for every real number y with y != 0, we have y3 != 0. That is, it states "The cube of every nonzero real number is nonzero." Note that this statement is equivalent to ∀y(y != 0 → y3 != 0). Finally, the statement ∃z > 0 (z2 = 2) states that there exists a real number z with z > 0 such that z2 = 2. That is, it states "There is a positive square root of 2." This statement is equivalent to ∃z(z > 0 ∧ z2 = 2).

What does the statement ∀xN(x) mean if N(x) is "Computer x is connected to the network" and the domain consists of all computers on campus?

The statement ∀xN(x) means that for every computer x on campus, that computer x is connected to the network. This statement can be expressed in English as "Every computer on campus is connected to the network."

Negating Quantified Expressions ¬∀xP(x) ≡ ∃x ¬P(x).

To show that ¬∀xP(x) and ∃xP(x) are logically equivalent no matter what the propositional function P(x) is and what the domain is, first note that ¬∀xP(x) is true if and only if ∀xP(x) is false. Next, note that ∀xP(x) is false if and only if there is an element x in the domain for which P(x) is false. This holds if and only if there is an element x in the domain for which ¬P(x) is true. Finally, note that there is an element x in the domain for which ¬P(x) is true if and only if ∃x ¬P(x) is true. Putting these steps together, we can conclude that ¬∀xP(x) is true if and only if ∃x ¬P(x) is true. It follows that ¬∀xP(x) and ∃x ¬P(x) are logically equivalent.

Let A(x) denote the statement "Computer x is under attack by an intruder." Suppose that of the computers on campus, only CS2 and MATH1 are currently under attack by intruders. What are truth values of A(CS1), A(CS2), and A(MATH1)?

We obtain the statement A(CS1) by setting x = CS1 in the statement "Computer x is under attack by an intruder." Because CS1 is not on the list of computers currently under attack, we conclude that A(CS1) is false. Similarly, because CS2 and MATH1 are on the list of computers under attack, we know that A(CS2) and A(MATH1) are true.

Statements involving predicates and quantifiers are logically equivalent This logical equivalence shows that we can distribute a universal quantifier over a conjunction. Furthermore, we can also distribute an existential quantifier over a disjunction. However, we cannot distribute a universal quantifier over a disjunction, nor can we distribute an existential quantifier over a conjunction.

if and only if they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for the variables in these propositional functions. We use the notation S ≡ T to indicate that two statements S and T involving predicates and quantifiers are logically equivalent.

predicate

refers to a property that the subject of the statement can have