Predicates and Quantifiers. Niloufar Shafiei
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1 Predicates and Quantifiers Niloufar Shafiei
2 Review Proposition: 1. It is a sentence that declares a fact. 2. It is either true or false, but not both. Examples: = 3. True Proposition Toronto is the capital of Canada. x + 1 = 2. False Proposition Neither true nor false 1
3 Review Logical Operators Negation p not p. Conjunction pq p and q. Disjunction pq p or q. Exclusive or pq p or q, but not both. Conditional statement pq If p, then q. Biconditional statement pq p if and only if q. 2
4 Predicate Logic More powerful Express a wide range of statements in mathematics and computer science 3
5 Predicates Variable: subject of the statement x > 3 Predicate: property that the subject of the statement can have 4
6 Predicates x > 3 denotes predicate P(x) value of propositional function P at x denotes variable 5
7 Predicates (example) P(x) : x>3. What are the truth value of P(4) and P(2)? Set variable x x=4 P(4): 4>3 True x=2 P(2): 2>3 False 6
8 Predicates (example) Q(x,y) : x = y+3. What are the truth value of Q(1,2) and Q(3,0)? Set variables x and y x=1, y=2 Q(1,2): 1=2+3 False x=3, y=0 Q(3,0): 3=0+3 True 7
9 Predicates (example) A(c,n) : Computer c is connected to network n. Assume computer CS1 is connected to network AirYork, but not to network Indigo. What are the truth value of A(CS1,AirYork) and A(CS1,Indigo)? Set variables c and n A(CS1,AirYork): Computer CS1 is connected to network AirYork. True A(CS1,Indigo): Computer CS1 is connected to network Indigo. False 8
10 Verification of computer programs Interchange the values of two variables x and y. 1. Temp := x 2. x := y 3. y := temp Precondition: conditions that input should satisfy P(x,y): x=a, y=b Postcondition: conditions that output should satisfy Q(x,y): x=b, y=a 9
11 Verification of computer programs Interchange the values of two variables x and y. 1. Temp := x 2. x := y 3. y := temp How to verify the program? Assume precondition P(x,y) holds P(x,y): x=a, y=b True Check the values of variables after each step of the program 1. Temp := x x=a, y=b, temp=a 2. x := y x=b, y=b, temp=a 3. y := temp x=b, y=a, temp=a At the end, check the postcondition Q(x,y) Q(x,y): x=b, y=a True 10
12 Review Propositional function P(x 1, x 2,, x n ) Assign a values to variables and form proposition with certain truth value Q(x,y) : x = y+3 Q(3,0) : 3 =
13 Quantifiers Create a proposition from a propositional function using Quantifiers Quantifiers express the range of elements the statement is about. The universal quantifier The existential quantifier 12
14 The universal quantifier The universal quantifier is used to assert a property of all values of a variable in a particular domain. 13
15 The universal quantifier The universal quantification of P(x) is P(x) for all values of x in the domain., denoted by x P(x) The universal quantifier For all For every For each All of For arbitrary The universal quantifier 14
16 The universal quantifier (example) P(x): x+1 > x The universal quantifier of P(x) is in the domain of real numbers: x P(x) (x is a real number) x ( x+1 > x ) (x is a real number) 15
17 The universal quantifier x P(x) When true? P(x) is true for every x in the domain When false? P(x) is not always true when x is in the domain (find a value of x that P(x) is false) 16
18 The universal quantifier An element for which P(x) is false is called a counterexample of x P(x). P(x): x>3 P(2): 2>3 is a counterexample of x P(x) 17
19 The universal quantifier (example) P(x): x+1 > x. What is the truth value of x P(x) in the domain of real numbers? Check if P(x) is true for all real numbers x+1 > x is true for all real number So, the truth value of x P(x) is true. 18
20 The universal quantifier (example) Q(x): x < 2. What is the truth value of x Q(x) in the domain of real numbers? Find a counterexample for x Q(x) Q(3): 3<2 is false x=3 is a counterexample for x Q(x), so x Q(x) is false. 19
21 The universal quantifier (example) P(x): x 2 > 0. What is the truth value of x P(x) in the domain of integers? Find a counterexample for x P(x) P(0): 0>0 is false x=0 is a counterexample for x P(x), so x P(x) is false. 20
22 The universal quantifier (example) A(x): Computer x is connected to the network. What is x A(x) in the domain of all computers on campus? x A(x) : For every computer x on campus, computer x is connected to the network. Every computer on campus is connected to the network. 21
23 The universal quantifier (example) P(x): x 2 x. What is the truth value of x P(x) in the domain of all real numbers? Find a counterexample for x P(x) P(1/2): 1/4 1/2 is false x=1/2 is a counterexample for x P(x), so x P(x) is false. 22
24 The universal quantifier (example) P(x): x 2 x. What is the truth value of x P(x) in the domain of all real numbers? How to find a counterexample? x 2 x. (x 2 - x) = x(x - 1) 0. x and (x-1) must both be zero or positive. x 0 and (x - 1) 0 x 0 and x 1 x 1 OR x and (x-1) must both be zero or negative. x 0 and (x - 1) 0 x 0 and x 1 x 0 0 < x < 1 such as x=1/2 is a counterexample 23
25 The universal quantifier (example) P(x): x 2 x. What is the truth value of x P(x) in the domain of all integers? Check if P(x) is true for all integers P(x) is true when x 1 or x 0. There is no integer between 0 < x < 1. So, x P(x) is true for the domain of all integers. 24
26 The universal quantifier x P(x) in the domain D If D can be listed as x 1, x 2,, x n. x P(x) in the domain D is the same as P(x 1 ) P(x 2 ) P(x n ) 25
27 The universal quantifier (example) P(x): x 2 < 10. What is the truth value of x P(x) in the domain of positive integers not exceeding 4? List the domain Domain is 1, 2, 3, 4. Find the equivalent conjunction and its truth value P(1) P(2) P(3) P(4) T T T F which is false So, P(4) is a counterexample and x P(x) is false. 26
28 The existential quantifier The existential quantifier is used to assert a property of at least one value of a variable in a domain. 27
29 The existential quantifier The existential quantification of P(x) is There exists an element x in the domain such that P(x)., denoted by x P(x) The existential quantifier The existential quantifier There exists There is For some For at least one 28
30 The existential quantifier (example) P(x): x > 3 The existential quantifier of P(x) is in the domain of integers: x P(x) (x is an integer) x ( x > 3 ) (x is an integer) 29
31 The existential quantifier x P(x) When true? There is an x for which P(x) is true. (find a value of x that P(x) is true.) When false? P(x) is false for every x. 30
32 The existential quantifier (example) P(x): x > 3. What is the truth value of x P(x) in the domain of real numbers? Check if P(x) is true for some real numbers x > 3 is true when x = 4. So, the truth value of x P(x) is true. 31
33 The existential quantifier (example) Q(x): x = x+1. What is the truth value of x Q(x) in the domain of real numbers? Check if Q(x) is false for all real numbers x = x+1 is false for all real numbers. So, the truth value of x Q(x) is false. 32
34 The existential quantifier x P(x) in the domain D If D can be listed as x 1, x 2,, x n. x P(x) in the domain D is the same as P(x 1 ) P(x 2 ) P(x n ) 33
35 The existential quantifier (example) P(x): x 2 > 10. What is the truth value of x P(x) in the domain of positive integers not exceeding 4? List the domain Domain is 1, 2, 3, 4. Find the equivalent disjunction and its truth value P(1) P(2) P(3) P(4) F F F T which is true. So, x P(x) is true. 34
36 Quantifiers (review) Statement x P(x) x P(x) When True? P(x) is true for every x. There is an x for which P(x) is true. When False? There is an x for which P(x) is false. P(x) is false for every x. 35
37 Translating from English into logical expression (example) Express the following statement using predicates and quantifiers? Every student in this class has studied calculus. Determine individual propositional function P(x): x has studied calculus. Translate the sentence into logical expression x P(x) domain: students in class 36
38 Translating from English into logical expression (example) Express the following statement using predicates and quantifiers? Some student in this class has visited Mexico. Determine individual propositional function P(x): x has visited Mexico. Translate the sentence into logical expression x P(x) domain: students in class 37
39 Translating from English into logical expression (example) Express the following statement using predicates and quantifiers? Every student in this class has visited either the US or Mexico. Determine individual propositional functions P(x): x has visited the US. Q(x): x has visited Mexico. Translate the sentence into logical expression x ( P(x) Q(x)) domain: students in class 38
40 Quantifiers with restricted domain Sometimes a variable in the domain must satisfy a condition. 39
41 Quantifiers with restricted domain (example) x < 0 (x 2 > 0). What does this statement mean in the domain of real numbers? (express it in English and logic using conditional statement) Express the statement in English For every real number x with x < 0, x 2 > 0. The square of a negative real number is positive. Find the equivalent if statement (find condition and property) Condition: x < 0 property: x 2 > 0 x ( x< 0 x 2 > 0 ) 40
42 Quantifiers with restricted domain (example) y 0 (y 3 0). What does this statement mean in the domain of real numbers? (express it in English and logic using conditional statement) Express the statement in English For every real number y with y 0, y 3 0. The cube of every nonzero real number is nonzero. Find the equivalent if statement (find condition and property) Condition: y 0 property: y 3 0 y ( y 0 y 3 0 ) 41
43 Quantifiers with restricted domain (example) z > 0 (z 2 = 2). What does this statement mean in the domain of real numbers? (express it in English and logic using conjunction) Express the statement in English There exists a real number z with z > 0, z 2 = 2. There is a positive square root of 2. Find the equivalent conjunction (find conditions) Conditions: z > 0 z 2 = 0 z ( z > 0 z 2 = 0 ) 42
44 Example of system specifications Express the following system specifications. Every mail message larger than one megabyte will be compressed. If a user is active, at least one network link will be available. Determine individual predicates Mail message is larger than one megabyte. Mail message will be compressed. User is active. Network link will be available. 43
45 Example of system specifications Every mail message larger than one megabyte will be compressed. If a user is active, at least one network link will be available. Introduce variables for each sentence Mail message is larger than one megabyte. P(m,x): Mail message m is larger than x megabytes. domain of m: all mail messages domain of x: positive real numbers Mail message will be compressed. Q(m): Mail message m will be compressed. domain of m: all mail messages 44
46 Example of system specifications Every mail message larger than one megabyte will be compressed. If a user is active, at least one network link will be available. Introduce variables for each sentence User is active. R(u): User u is active. domain of u: all users Network link will be available. S(n): Network link n is available. domain of n: all network links 45
47 Example of system specifications Every mail message larger than one megabyte will be compressed. If a user is active, at least one network link will be available. Translate each specification into logical expression Every mail message larger than one megabyte will be compressed. P(m,x): Mail message m is larger than x megabytes. Q(m): Mail message m will be compressed. m (P(m,1) Q(m)) 46
48 Example of system specifications Every mail message larger than one megabyte will be compressed. If a user is active, at least one network link will be available. Translate each specification into logical expression If a user is active, at least one network link will be available. R(u): User u is active. S(n): Network link n is available. u R(u) n S(n) 47
49 Example Express the following sentences using predicates and quantifiers. All lions are fierce. Some lions do not drink coffee. Assume P(x): x is a lion. Q(x): x is fierce. R(x): x drinks coffee. Translate each sentence into logical expression All lions are fierce. x (P(x) Q(x)) 48
50 Example Express the following sentences using predicates and quantifiers. All lions are fierce. Some lions do not drink coffee. Assume P(x): x is a lion. Q(x): x is fierce. R(x): x drinks coffee. Translate each sentence into logical expression Some lions do not drink coffee. x (P(x) R(x)) 49
51 Precedence of quantifiers The quantifiers and has higher precedence than all logical operators. Example: x p(x) Q(x) (x p(x)) Q(x) Operators Precedence
52 Binding variables Variable Bound Quantifiers Free Not bound Turn a propositional function into a proposition All variables must be bound. 51
53 Binding variables (example) x (x+y=1). Is it a proposition? Check if any variable is free Variable x bound Variable y Free Since variable y is free, it is not a proposition. 52
54 Logical equivalences Assume S and T are two statements involving predicates and quantifiers. S and T are logically equivalent if and only if they have the same truth value no matter which predicates are substituted and which domain is used for the variables, denoted by S T. 53
55 Logical equivalences (example) Show that x (P(x) Q(x)) and x P(x) x Q(x) are logically equivalent. Part1: Show if x (P(x) Q(x)) is true then x P(x) x Q(x) is true. (using direct technique) Assume x (P(x) Q(x)) is true. If a is in the domain, then P(a) Q(a) is true. So, P(a) is true and Q(a) is true. Since P(a) and Q(a) are both true for every element in the domain, x P(x) and x Q(x) are both true. So, x P(x) x Q(x) is true. 54
56 Logical equivalences (example) Show that x (P(x) Q(x)) and x P(x) x Q(x) are logically equivalent. Part2: Show if x P(x) x Q(x) is true then x (P(x) Q(x)) is true. (using direct technique) Assume x P(x) x Q(x) is true. So, x P(x) is true and x Q(x) is true. If a is in the domain, then P(a) is true and Q(a) is true. If P(a) is true and Q(a) is true, then P(a) Q(a) is true. Since P(a) Q(a) is true for every element in the domain, x (P(x) Q(x)) is true. So, x (P(x) Q(x)) x P(x) x Q(x). 55
57 Logical equivalences (example) Show that x (P(x) Q(x)) and x P(x) x Q(x) are not logically equivalent. Give an example that x (P(x) Q(x)) and x P(x) x Q(x) have different truth values. P(x): x is odd. Q(x): x is even. (in the domain of integers.) For all element (P(x) Q(x)) is true. (all x is odd or even.) So, x (P(x) Q(x)) is true. For all element P(x) is false. (all x is not odd.) For all element Q(x) is false. (all x is not even.) So, x P(x) x Q(x) is false. Thus, x (P(x) Q(x)) and x P(x) x Q(x) are not logically equivalent. 56
58 Negation (review) Let p be a proposition. The negation of p, denoted by p, is the proposition It is not the case that p. 57
59 Negating quantified expression Statement x P(x) x P(x) When False? There is an x for which P(x) is false. P(x) is false for every x. x P(x) x P(x) x P(x) x P(x) 58
60 Negating quantified expression Assume x P(x) is: Every student has taken a course in calculus. (x P(x)) is: It is not the case that every student has taken a course in calculus. There is a student who has not taken a course in calculus. x P(x) (x P(x)) x P(x) 59
61 Negating quantified expression Show that (x P(x)) and x P(x) are logically equivalent. Show x P(x) is true if and only if x P(x) is true. (using direct technique) x P(x) is true if and only if x P(x) is false. x P(x) is false if and only if there is an element in the domain for which P(x) is false. There is an element for which P(x) is false if and only if there is an element for which P(x) is true. There is an element for which P(x) is true if and only if x P(x) is true. Thus, (x P(x)) and x P(x) are logically equivalent. 60
62 Negating quantified expression Assume x P(x) is: There is a student who has taken a course in calculus. (x P(x)) is: It is not the case that there is a student who has taken a course in calculus. Every student has not taken a course in calculus. x P(x) (x P(x)) x P(x) 61
63 Negating quantified expression Show that (x P(x)) and x P(x) are logically equivalent. Show x P(x) is true if and only if x P(x) is true. (using direct technique) x P(x) is true if and only if x P(x) is false. x P(x) is false if and only if there is no element in the domain for which P(x) is true. There is no element for which P(x) is true if and only if for all elements P(x) is true. For all elements P(x) is true if and only if x P(x) is true. Thus, (x P(x)) and x P(x) are logically equivalent. 62
64 De Morgan s laws for quantifiers Negation Equivalent st. When true? When false? x P(x) x P(x) There is an x that P(x) is false. For all x P(x) is true. x P(x) x P(x) For all x P(x) is false. There is an x that P(x) is true. 63
65 De Morgan s laws for quantifiers x P(x) in the domain D If D can be listed as x 1, x 2,, x n. x P(x) P(x 1 ) P(x 2 ) P(x n ) x P(x) (P(x 1 ) P(x 2 ) P(x n )) P(x 1 ) P(x 2 ) P(x n )) 64
66 De Morgan s laws for quantifiers x P(x) in the domain D If D can be listed as x 1, x 2,, x n. x P(x) P(x 1 ) P(x 2 ) P(x n ) x P(x) (P(x 1 ) P(x 2 ) P(x n )) P(x 1 ) P(x 2 ) P(x n )) 65
67 Negating quantified expression (example) What is the negation of the statement All Canadians eat sushi? Determine individual propositional function P(x): x eats sushi. Then translate the sentence into logical expression x P(x) domain: Canadians Find the negation of x P(x) x P(x) domain: Canadians Translate x P(x) into English sentence Some Canadian does not eat sushi. 66
68 Negating quantified expression (example) What is the negation of the statement There is an honest politician? Determine individual propositional function P(x): x is an honest politician. Then translate the sentence into logical expression x P(x) domain: politicians Find the negation of x P(x) x P(x) domain: politicians Translate x P(x) into English sentence Every politician is dishonest. 67
69 Negating quantified expression (example) What is the negation of x (x 2 > x)? ( x (x 2 > x) ) x (x 2 > x) x (x 2 x) 68
70 Negating quantified expression (example) What is the negation of x (x 2 = 2)? (x (x 2 = 2) ) x (x 2 = 2) x (x 2 2) 69
71 Recommended exercises 5,10,13,19,21,23,25,33,35,39,61 70
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