The Foundations: Logic and Proofs

The Foundations: Logic and Proofs Predicates and Quantifiers

Predicates “x is greater than 3” • This statement is neither true nor false when the value of the variable is not specified • This statement has two parts: • The first part (subject) is the variable x. • The second (predicate) is “is greater than 3”.

Predicates • We can denote this statement by P(x), where P denotes the predicate “is greater than 3”. Once a value has been assigned to x, the statement P(x) becomes a proposition and has a truth values • P(x) is called Proposition function P at x • P(x1,x2,x3,………,xn). • P is called n-place or (n-ary) predicate.

Predicates • Examples: • Let P(x) denote “ x is greater than 3”. What are the truth values of P(4) and P(2)? • Let Q(x, y) denote “x=y+3”. What are the truth values of Q(1,2) and Q(3,0)? • Let A(c, n) denote “computer c is connected to network n”, suppose that the computer MATH1 is connected to network CAMPUS2, but not to network CAMPUS1, What are the truth values of A(MATH1, CAMPUS1) and A(MATH1, CAMPUS2)?

Quantifiers Universal quantification • Which tell us that a predicate is true for every element under consideration( Domain / Discourse). • The universal quantificationof P(x) is the statement “P(x) for all values of x in the domain” • x P(x) read as “for all x P(x)” or “for every x P(x)” •  is called universal quantifier

Quantifiers Existential quantification • Which tell us that there is one or more element under consideration for which the predicate is true. • The existential quantificationof P(x) is the statement “there exists an element x in the domain such that P(x)” • x P(x) read as “there is an x such that P(x)” or “there is at least one x such that P(x)” or “for some x P(x)” •  is called existential quantifier

Quantifiers • The area of logic that deals with predicates and quantifiers is called predicate calculus. • x P(x) is • True when: P(x) is true for every x • False when: There is an x for which P(x) is false • x P(x) is • True when: There is an x for which P(x) is true • False when: P(x) is false for every x

Quantifiers • Examples: • Let Q(x) “x<2” . What is the truth value of x Q(x) when the domain consists of all real numbers? • Q(x) is not true for every real number x, for example Q(3) is false • x =3 is a counterexample for the statement x Q(x) Thus x Q(x) is false

Quantifiers • Examples: • What is the truth value of x (x2 x) when the domain consists of: • a) all real number? B) all integers? • a ) is false because (0.5)2 0.5 , x2 x is false for all real numbers in the range 0<x<1 • b) is true because there are no integer x with 0<x<1

Quantifiers • Examples: • Let Q(x) “x>3”. What is the truth value of x Q(x) when the domain consists of all real numbers? • Q(x) is sometimes true , for example Q(4) is true • Thus x Q(x) is true

Quantifiers • Examples: • Let Q(x) “x=x+1”. What is the truth value of x Q(x) when the domain consists of all real numbers? • Q(x) is false for every real number • Thus x Q(x) is false

Quantifiers • Note that : x Q(x) is false if there is no elements in the domain for which Q(x) is true or the domain is empty. • When all the elements in the domain can be listed x1, x2, x3, x4, ……., xn : • x Q(x) is the same as the conjunction • Q(x1)  Q(x2) ….  Q(xn) • x Q(x) is the same as the disjunction • Q(x1)  Q(x2) ….  Q(xn) • Precedence of quantifiers •  and  have higher precedence than all logical operators 

Quantifiers • Examples: • Let Q(x) “x2<10”. What is the truth value of x Q(x) when the domain consists of the positive integers not exceeding 4? • x Q(x) is the same as the conjunction • Q(1)  Q(2)  Q(3)  Q(4). • Q(4) is false. Thus x Q(x) is false.

Quantifiers • Examples: • Let Q(x) “x2<10”. What is the truth value of x Q(x) when the domain consists of the positive integers not exceeding 4? • x Q(x) is the same as the disjunction • Q(1)  Q(2)  Q(3)  Q(4). • Q(4) is false. Thus x Q(x) is false

Quantifiers • If domain consists of n (finite) object and we need to determine the truth value of. • x Q(x) • Loop through all n values of x to see if Q(x) is always true • If you encounter a value x for which Q(x) is false, exit the loop with x Q(x) is false • Otherwise x Q(x) is true • x Q(x) • Loop through all n values of x to see if Q(x) is true • If you encounter a value x for which Q(x) is true, exit the loop with x Q(x) is true • Otherwise x Q(x) is false

Quantifiers with restricted domain • The restriction of a universal quantification is the same as the universal quantification of a conditional statement • x<0 (x2>0) same as x (x<0  x2>0) “The square of a negative real number is positive” • The restriction of a existential quantification is the same as the existential quantification of a conjunction • z >0 (z2=2) same as z(z>0  z2=2) • “There is a positive square root of 2”

Binding variables • When a quantifier is used on the variable x , we say that this occurrence of the variable is bound. • x (x+y=1) • The variable x is bounded by the existential • quantification x and the variable y is free • All variable that occur in a propositional function must be bound or equal to particular value to turn it into proposition.

Binding variables • Examples: • x (P(x)  Q(x)) x R(x) • All variables are bounded • The scope of • the first quantifier x is the expression P(x)Q(x), • second quantifier x is the expression R(x) • Existential quantifier binds the variable x in P(x)Q(x) • Universal quantifier binds the variable x in R(x)

Quantifiers • Other Quantifiers Uniqueness Quantifier ! or 1 “!x P(x)” = “There exists a unique x such that P(x) is true” 19

Negating Quantified Expressions • “Every student in this class has taken a calculus course” xP(x) where • P(x) is “x has taken a calculus course” • Domain = “Students in class” • Negation is “There is a student that has not taken a calculus course” x P(x) xP(x)  xP(x)

Negating Quantified Expressions • If domain of P(x) consists of n elements “x1,x2,x3,…,xn” then xP(x)  (P(x1)P(x2)P(x3)…P(xn)) by DeMorgan’s laws  P(x1) P(x2) P(x3)… P(xn)  xP(x) • De Morgan’s Lwas for Quantifiers xP(x)  xP(x)  xP(x)  xP(x)

Nested Quantifiers • Two quantifiers are nested if one is within the scope of another e.g.x y (x+y=0) • Examples: • xy(x+y=y+x) • xy((x>0)(y<0)(xy<0))

Order of Nested Quantifiers • Order of nested quantifiers is important if they are different. • Example: • If Q(x,y) denotes x+y=0, what are the truth values of quantifications • xy Q(x,y) • yx Q(x,y)

Negating Nested Quantifiers • Example: Express the negation of xy(xy=1) xy(xy=1) by DeMorgan’s laws xy(xy=1) xy(xy=1) xy(xy1)

Negating Nested Quantifiers • Example: Express the statement (There does not exist a woman who has taken a flight on every airline in the world) Let P(w,f) be “w has taken f” and Q(f,a) be “f is a flight on a”, then w a f(P(w,f)Q(f,a)) w a f(P(w,f)Q(f,a)) w a f(P(w,f)Q(f,a)) w a f (P(w,f)Q(f,a)) w a f ( P(w,f)Q(f,a))

The Foundations: Logic and Proofs