1. 1 Introduction to Abstract Mathematics Chapter 3: The Logic of Quantified Statements. Predicate Calculus Instructor: Hayk Melikya melikyan@nccu.edu 3.1 3.2 Predicates and Quantified Statements

2. 2 Introduction to Abstract Mathematics Predicates (open sentences) A predicate (an open sentence) is a sentence that contains a finite number of variables and becomes a proposition ( or statement) when specific values are substituted for the variables x + 2 = y Example: P (x, y) = x = 1 and y = 3: P(1, 3) is true x = 1 and y = 4: P(1,4) is false What about P(3, 5) ? Q(x, y) := “ if x and y are integers then x + 3y is multiple of 5 ” ( substitute x and y by any integer ) Q(2, 1):= “ if 2 and 1 are integers then 2 + 3 is multiple of 5 ”

3. 3 Introduction to Abstract Mathematics Domain of Predicate( Universe ) The domain of a predicate variable is a set of all values that may be substituted in place of the variable If A is a set then P(x) := “ x is an element of A” is a predicate which is denoted x  A. The negation of P(x), ~P(x) := “ x is not an element of A” will be denoted x  A.

4. 4 Introduction to Abstract Mathematics Sets R Set of all real numbers Z Set of all integers Q Set of all rational numbers Sets can be defined directly by specifying its elements between a pair of braces. A = {1, 2, 4, 8, 16, 32}, or C = {COMP2200, MATHC1100, HIST2100} 4  A is true, 5  A is false, 5  A is true You should remember the symbols used to denote for each set such as increasing collection of set Z, Q, R since we will be referring to these sets in the remainder of the book.

5. 5 Introduction to Abstract Mathematics Truth Set of Predicates If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements in D that make P(x) true proposition when substituted for x. The truth set is denoted as: {x  D | P(x)} which is read ” the set of all x in D such that P(x) is true” Example: Let P(x) := “ x 2 + 3x – 4 = 0 ” with the universe R (real numbers) What is the truth set for P(x) ? {x  R | x 2 + 3x – 4 = 0} = {1, -4 }.

6. 6 Introduction to Abstract Mathematics Universal Quantifier Let P(x) be a predicate with domain D. A universal statement (proposition) is a statement (proposition) in the form “  x  D, P(x) ”. It is true iff P(x) is true for every x from D. It is false iff P(x) is false for at least one x from D. A value of x form which P(x) is false is called a counterexample to the universal statement. Examples: D = {1, 2, 3, 4, 5}: and P(x):= “x² ≥ x ” The universal statement  x  D, P(x) is true (why ?) Method of exhaustion: Check for P(1), P(2), P(3), P(4), P(5) What about  x  R, P(x) (true or false?) ( can you check all rationals?) What do you think about P(0.4) ? The symbol  is very popular symbol in mathematics in formal and informal sense and is called universal quantifier. The symbol  denotes “ for all” or “for every”

7. 7 Introduction to Abstract Mathematics Existential Quantifier Let P(x) be a predicate with domain D. An existential statement (proposition) is a statement (proposition) in the form “  x  D, P(x)”. It is true iff P(x) is true for at least one x from D. It is false iff P(x) is false for every x from D. Examples: Let D = {5, 6, 7, 8, 9} and P(x):= “ x² = x ” What we can say about  x  D, P(x) What about  x  Z, P(x ) The symbol  denotes “ there exists” and is called existential quantifier. Note:1. If proposition  x  D, P(x) is true then the truth set of P(x) is the universe D: D = {x  D | P(x)} 2. If proposition  x  D, P(x) is true then the truth set P(x) not empty : {x  D | P(x) } ≠ Ø

8. 8 Introduction to Abstract Mathematics Universal Conditional Statement Universal conditional statement “  x, if P(x) then Q(x)”: or  x  U, P(x)  Q(x) –  x  R, if x > 2, then x 2 > 4 Empty domains: all pink elephants speak Latin Universal conditional statement is called vacuously true or true by default iff P(x) is false for every x in D

9. 9 Introduction to Abstract Mathematics Margin Note: In books when one sees a statement like “If x is an integer then x is a rational number,” one means (  x  Z)(x  Z   x  Q) or (  x  Z)(x  Q) In other words the universal quantifier is understood. Existential quantifier is always explicitly be present to mean that.

10. 10 Introduction to Abstract Mathematics Negation of Quantified Statements The negation of a universally quantified statement  x  D, P(x) is  x  D, ~P(x) The negation of an existentially quantified statement  x  D, P(x) is  x  D, ~P(x) The negation of a universal conditional statement  x  D, P(x)  Q(x) is  x  D, P(x)  ~Q(x)

11. 11 Introduction to Abstract Mathematics Exercises Write negations for each of the following statements: –All dinosaurs are extinct –No irrational numbers are integers –Some exercises have answers –All COBOL programs have at least 20 lines –The sum of any two even integers is even –The square of any even integer is even

12. 12 Introduction to Abstract Mathematics Below are English language interpretations of predicate logic sentences. Some sentences include more than one quantifier.

13. 13 Introduction to Abstract Mathematics Multiple Quantified Statements v For all positive numbers x, there exists number y such that y < x v There exists number x such that for all positive numbers y, y < x v For all people x there exists person y such that x loves y v There exists person x such that for all people y, x loves y v Definition of mathematical limit

14. 14 Introduction to Abstract Mathematics Negation of Multiple Quantified Statements v The negation of  x,  y, P(x, y) is logically equivalent to  x,  y, ~P(x, y) v The negation of  x,  y, P(x, y) is logically equivalent to  x,  y, ~P(x, y)

15. 15 Introduction to Abstract Mathematics Necessary and Sufficient Conditions, Only If v  x, r(x) is a sufficient condition for s(x) means:  x, if r(x) then s(x) v  x, r(x) is a necessary condition for s(x) means:  x, if s(x) then r(x) v  x, r(x) only if s(x) means:  x, if r(x) then s(x)

16. 16 Introduction to Abstract Mathematics Exercises Rewrite  !x  D, P(x) without using the symbol  ! Determine whether a pair of quantified statements have the same truth values –  x  D, (P(x)  Q(x)) vs (  x  D, P(x))  (  x  D, Q(x)) –  x  D, (P(x)  Q(x)) vs (  x  D, P(x))  (  x  D, Q(x)) –  x  D, (P(x)  Q(x)) vs (  x  D, P(x))  (  x  D, Q(x)) –  x  D, (P(x)  Q(x)) vs (  x  D, P(x))  (  x  D, Q(x)) Let P(x) and Q(x) be predicates with the common domain D. P(x)  Q(x) means that every element in the truth set of P(x) is in the truth set of Q(x). P(x)  Q(x) means that P(x) and Q(x) have identical truth sets

17. 17 Introduction to Abstract Mathematics Interchanging Quantifiers   Does the order of the quantifiers make a difference in the The four drawings a), b), c) and d) in Figure illustrate visually the following implication (  x)(  y)P(x,y)  (  y)(  x)P(x,y)  (  x)(  y)P(x,y)  (  x)(  y)P(x,y)

18. 18 Introduction to Abstract Mathematics (  x)(  y)P(x, y)  (  y)(  x)P(x, y)  (  x)(  y)P(x, y)  (  x)(  y)P(x, y)

19. 19 Introduction to Abstract Mathematics If the proposition (  x)(  y)P(x, y) true then for all x and y, the Predicate P(x, y) is true. That is, the statement is true if P(x, y) is true everywhere in the first quadrant, which we have shaded. The theorem (  y)(  x)P(x, y) means there exists a y, say y 0, such that for all y the statement P(x, y 0 ) is true. We draw the horizontal line x = y 0 illustrating that the theorem is true if P(x,y) is true everywhere on this line. If we permute the quantifiers the theorem becomes (  x)(  y)P(x,y) which says for all x there exists a y = f (x) such that P(x,y) is true. Note that the constant function in b), being a special case of the arbitrary function f(x) in c) shows the important implication (  y)(  x)P(x, y)  (  x)(  y)P(x, y )

20. 20 Introduction to Abstract Mathematics The statement (  y)(  y)P(x, y) is a pure existence theorem and is true if there exists at least one point ( x 0, y 0 ) where P( x, y) is true. This statement is the weakest of the four statement (  x)(  y)P(x, y)  (  y)(  y)P(x,y) The implication does not go the other way as proven by the following counterexample ( P(x, y):= “ x < y ” ) (  x)(  y)(x< y)  (  x)(  y)(x <y) Here the hypothesis is true but the conclusion is false.

21. 21 Introduction to Abstract Mathematics Equivalence of two predicates Two predicated P(x) and Q(x) with the specified universe U are said to be equivalent over the universe U of they have same truth set. We will write P(x)  U Q (x) if P(x) and Q(x) are equivalent of the universe U. Two quantified predicates are said to be equivalent if they are equivalent over the any universe. Example: (  x) (x > 3)  Z (  x) (x > 3.7 ) But it is not true (  x) (x > 3)  R (  x) (x > 3.7 ). Compare their truth set. Example: If P(x) and Q(x) are predicates then (  x) (P(x)  Q(x)) and (  x)(Q(x))  P(x)) are equivalent over any universe (  x) (P(x)  Q(x))  (  x)(Q(x)  P(x) )

22. 22 Introduction to Abstract Mathematics Negating quantified predicates: Theorem: Let P(x) be a predicate with variable x then  (  x)P(x)  (  x)  P(x)  (  x)P(x)  (  x)  P(x) Proof. If U is the universe, then  (  x) (P(x) is true in U if (  x)(P(x) is false in U which means that the truth set of it is not the univese or for some a from universe P(a) is not true hance  P(a) is true which tells us that (  x)  P(x) true. This theorem is very useful for finding denials of quantified sentences

23. 23 Introduction to Abstract Mathematics More Examples The following table shows how statements in predicate logicare negated.

24. 24 Introduction to Abstract Mathematics Practice problems 1. Study the Sections 3.1 and 3.2 from your textbook. 2. Be sure that you understand all the examples discussed in class and in textbook. 3. Do the following problems from the textbook: Exercise 3.1 # 3, 5, 6, 7, 13, 16, 19, 25, 28, 29 Exercise 3.2 # 2, 4, 15, 18, 24.