Chapter 3 The Logic of Quantified Statements

Quantified Statements A quantified statement is a statement that specifies quantity (of something). Examples: There is at least one student absent today. Everybody in the class passes the 1st quiz. The equation x2 + 5x – 6 = 0 has exactly two different real solutions. A non-quantified statement: “3331 is a prime number.”

Quantified Statements Most interesting properties (theorems) in mathematics are quantified statements. Examples: All polynomial functions are differentiable. For all real number x, sin2x + cos2x = 1. For any whole number n > 1, there is always a prime number between n and 2n.

Quantifiers A quantifier is a symbols that expresses the quantity of a certain type of objects. Only two types of quantifiers will be needed. Universal quantifier:  x(x2  0) means “for all x, x2  0” Existential quantifier:  n (n×n = n+n) means “there exists an n such that n×n = n+n”

Quantifiers How do we express “there exists only one” with a quantified statement ? ! x P(x) means there is exactly one x such that P(x) is true. It is the abbrev. of  x { P(x)  [y (y  x  P(y))]}

Section 3.1 Predicate Calculus In a quantified statement, the sentence after the quantifier is called a predicate. Example: x such that x2 – 3x + 2 = 0 the sentence “x2 – 3x + 2 = 0” is called a predicate. A predicate is a sentence with a finite number of variables, and it normally does not have any truth value. A predicate becomes a statement when specific values are substituted for the variables. Examples: P(x): x2 – 3x + 2 = 0 Q(n): n is the sum of two integer squares R(x): sin2(x)  cos2(x)

Examples: P(x): x2 – 3x + 2 = 0 Q(n): n is the sum of two integer squares R(x): sin2(x)  cos2(x) Domain of a predicate is the set of values that can be substituted to the variable(s) in the predicate. dom P(x) = the set of real numbers dom Q(n) = the set of integers dom R(x) = the set of real numbers

Notations for some common sets R is the set of real numbers Z is the set of integers Zahlen (German for numbers). Q is the set of rational numbers N is the set of natural numbers C is the set of complex numbers Another notation: P(x)  Q(x) means that for any c in the common domain of P and Q, if P(c) is true then so is Q(c). Example: x > 4  x2 > 16 Note: A predicate can be interpreted as a function whose (output) value(s) is a proposition.

Quantifiers There are two ways to convert a predicate to a statement: assigning values to its variables quantifying its variables Examples For all real number x, sin2(x) + cos2(x) = 1 There is at least one positive integer n such that n×n = n + n Universal quantifier:  xD, P(x) means for every x in the domain D, P(x) Existential quantifier:  nD, Q(n) means there is at least one n in the domain D such that Q(n)

Bound and Free variables In the following predicate  xy P(x, y, z) x and y are called bound variables z is called a free variable. Free variables can be substituted by numbers but bound variables cannot.

Truth and Falsity of Quantified statements A universal statement xD P(x) is true if for each x in D, P(x) is true Example: The statement “n N (n is the sum of 4 integer squares)” is true if 0 is the sum of 4 integer squares, 1 is the sum of 4 integer squares, 2 is the sum of 4 integer squares, … … Note: the above example explains why predicate calculus is much more difficult than propositional calculus – the truth value of a quantified statement cannot be determined by a calculator or even computer.

A universal statement xD P(x) is false if there is at least one element w in D such that P(w) is false. Note: In some unusual cases, the domain D turns out to be empty, and therefore it is impossible to find any w in D such that P(w) is false. xD P(x) cannot be false, xD P(x) must be true. We say that xD P(x) is true by default Example: “x{x: sin(x)>1} ( x3 – 7x + 5 = 0 )” is true.

An existential statement xD Q(x) is true if there is a w in D such that Q(w) is true Example: The statement “ nN (n is the sum of its proper factors)” is true because 6 = 1 + 2 + 3 is the sum of all it’s proper factors. (note: proper here means less than n itself.) There are of course more numbers of this type (i.e. perfect numbers) but it is irrelevant to the truth of this statement. The next three perfect numbers are 28, 496, 8128.

Note: the truth of an existential statement cannot always be determined by a computer either, as the search may last for ever. Example: “ nN (n is odd  n is perfect)” is still an unanswered question. If such an odd perfect number does exist, it must be greater than 10200.

Negations of Quantified Statements [xD P(x) ] is equiv. to xD [P(x)] [xD P(x) ] is equiv. to xD [P(x)] Examples: Everyone in this class has a computer at home. negation: There is at least one person in this class that does not have a computer at home. There was at least one day last year that I did not watch TV. Everyday last year I watched TV (at least for a while).

Section 3.2 Multiple Quantifiers A statement with multiple quantifiers is more complex but is more expressive as well. Examples: “there is no largest number” can be rephrased as [ yR xR (x  y  y > x) ] “Every real number has an additive inverse” is expressed as xR yR ( x + y = 0 ) “limit of f (x) = L at x = c” is defined as  > 0   > 0 s.t. x (0<  x – c  <    f (x) – L  < ) “If n is an integer greater than 1, there is a prime number between n and 2n” n Z [n>1  p Z (“p is prime”  n < p < 2n) ]

Negations [  x D  y D P(x, y)]   x D  y D [P(x, y)] Remark: if there is no change to the domain, we can skip the domain when it is clear in the context. [ xy  z Q(x, y, z)]  x y  z [ Q(x, y, z)] [ xy  z Q(x, y, z)]  x y  z [ Q(x, y, z)]

Relations among ,  and ,  If the domain D of quantification is finite, say D = {x1, x2, … , xn} then a universal statement is just a conjunction x D Q(x)  Q(x1)  Q(x2)  …  Q(xn) an existential statement is just a disjunction x D Q(x)  Q(x1)  Q(x2)  …  Q(xn) If the domain D is infinite, then a universal statement is an “infinite conjunction” an existential statement is an “infinite disjunction” The above informal thought suggests that predicate calculus is an extension of propositional calculus.

Section 3.3 Arguments with quantified statements Universal Modus Ponens x { P(x)  Q(x) } P(a) for a particular a  Q(a) Example: For any integer n, if the sum of the digits in n is divisible by 3, then n is divisible by 3. The sum of digits in 54627 is divisible by 3.  54627 is divisible by 3.

Universal Modus Tollens x { P(x)  Q(x) } Q(a) for a particular a  P(a) Example: For any real sequence {an}, if an is convergent, then an goes to 0 as n goes to  . does not go to 0 as n goes to  is not convergent.