ICS 253: Discrete Structures I Spring Semester 2014 (2013-2) Predicates and Quantifiers Dr. Nasir Al-Darwish Computer Science Department King Fahd University of Petroleum and Minerals darwish@kfupm.edu.sa
Propositional Predicate • Definition:Apropositional predicate P(x) is a statement that has a variable x. • Examples of P(x) P(x) = “The Course x is difficult” P(x) = “x+2 < 5” • Note: a propositional predicate is not a proposition because it depends on the value of x. • Example: If P(x) = “x > 3”, then P(4) is true but not P(1). • A propositional predicate is also called a propositional function
Propositional Predicate – cont. • It also possible to have more than one variable in one predicate, e.g., Q(x,y) =“x > y-2” • P(x) is a function (a mapping) that takes a value for x and produce either true or false. • Example: P(x) = “x2 > 2” , P:SomeDomain {T, F} • Domain of x is called the domain (universe) of discourse
Quantification • A predicate (propositional function) could be made a proposition by either assigning values to the variables or by quantification. • Predicate Calculus: Is the area of logic concerned with predicates and quantifiers.
Quantifiers 1. Universal quantifier:P(x) is true for all (every) x in the domain. We writexP(x) 2. Existential quantifier:there exists at least one x in the domain such that P(x) is true. We write x P(x) 3. Others: there exists a unique x such that P(x) is true.We write !xP(x)
Universal Quantification • Uses the universal quantifier (for all) • xP(x) corresponds to “p(x) is true for all values of x (in some domain)” • Read it as “for all xp(x)” or “for every xp(x)” • Other expressions include “for each” , “all of”, “for arbitrary” , and “for any” (avoid this!) • A statement xP(x) is false if and only if p(x) is not always true (i.e., P(x) is false for at least one value of x) • An element for which p(x) is false is called acounterexampleof xP(x); one counterexample is all we need to establish that xP(x) is false
Universal Quantification - Examples • Example 1: Let P(x) be the statement “x + 1 > x” . What is the truth value of ∀x P(x), where the domain for x consists of all real numbers? • Solution: Because P(x) is true for all real numbers x, the universal quantification ∀x P(x) is true. • Example 2: Suppose that P(x) is “x2 > 0” . What is the truth value of ∀x P(x), where the domain consists of all integers. • Solution: We show ∀x P(x) is false by a counterexample. We see that x = 0 is a counterexample because for x = 0, x2 = 0, thus there is some integer x for which P(x) is false.
Existential Quantification • Uses the existential quantifier (there exists) • xP(x) corresponds to “There exists an element x (in some domain) such that p(x) is true” • In English, “there is”, “for at least one”, or “for some” • Read as “There is an x such that p(x)”, “There is at least one x such that p(x)”, or “For some x, p(x)” • A statement xP(x) is false if and only if “for all x, P(x) is false”
Existential Quantification - Examples • Example 1: Let P(x) denote the statement “x > 3”. What is the truth value of ∃xP(x), where the domain for x consists of all real numbers? • Solution: Because “x > 3” is true for some values of x , for example, x = 4, the existential quantification ∃x P(x) is true. • Example 2: Let Q(x) denote the statement “x = x + 1” . What is the truth value of ∃x Q(x), where the domain consists of all real numbers? • Solution: Because Q(x) is false for every real number x, the existential quantification ∃x Q(x) is false.
Predicates and Negations
Domain (or Universe) of Discourse • Cannot tell if a quantified predicate P(x) is true (or false) if the domain of x is not known. • The meaning of the quantified P(x) changes when we change the domain. • The domain must always be specified when universal or existential quantifiers are used; otherwise, the statement is ambiguous.
Quantification Examples • P(x) = “x+1 = 2”Domain is R (set of real numbers) Proposition Truth Value xP(x) x P(x) xP(x) x P(x) !xP(x) !x P(x) F F T T T F
Quantification Examples • P(x) = “x2 > 0” Domain Proposition Truth Value R xP(x) Z xP(x) Z - {0} xP(x) Z !x P(x) N={1,2, ..} xP(x) F F T T F
Quantification Examples Proposition Truth Value xR (x2 x) !xR (x2 < x) x(0,1) (x2 < x) x{0,1} (x2 = x) x P(x) F F T T T
Logically Equivalence Definition:Two statements involving predicates & quantifiers are logically equivalent if and only if they have the same truth values independent of the domains and the predicates. Examples: • ( xP(x) ) xP(x) • ( xP(x) ) xP(x)
Theorem If the domain of discourse is finite, say Domain = {x1, x2, …, xn}, then • xP(x) xP(x) P(x1) P(x2) ... P(xn) P(x1) P(x2) ... P(xn)
Precedence , , , , , Example: ((xP(x) ) ((!xQ(x) )(xP(x) )))
Correct Equivalences • x ( P(x) Q(x) ) xP(x) xQ(x) • This says that we can distribute a universal quantifier over a conjunction • x ( P(x) Q(x) ) xP(x) x Q(x) • This says that we can distribute an existential quantifier over a disjunction • The preceding equivalences can be easily proven if we assume a finite domain for x = {x1, x2, …, xn} • Note: we cannot distribute a universal quantifier over a disjunction, nor can we distribute an existential quantifier over a conjunction.
Wrong Equivalences • x ( P(x) Q(x) ) xP(x) xQ(x) • Read as “there exists an x for which both P(x) and Q(x) are true is equivalent to there exists an x for which P(x) is true and there exists an x for which Q(x) is true”. One can construct an example that makes the above equivalence false. Consider the following • x ( P(x) Q(x) ) = F but • xP(x) xQ(x) = T, since P(1) is true and Q(2) is true
Wrong Equivalences • x ( P(x) Q(x) ) xP(x) x Q(x) • One can construct an example that makes the above equivalence false. Consider the following • x ( P(x) Q(x) ) = T but • xP(x) xQ(x) = F F = F
Wrong Equivalences • ( xP(x) ) Q(x) x(P(x) Q(x) ) • Notice that the LHS = ( xP(x) ) Q(x)is not fully quantified. So it cannot be equivalent to RHS.
Quantifiers with restricted domains • What do the following statements mean for the domain of real numbers? Be careful about → and ˄ in these statements
