PredicatesQuantifiers.docx
- 1. 1/ 33 Introduction Propositional Functions Slides by Christopher M. Bourke Propositional Functions Instructor: Berthe Y. Choueiry Quantifiers Logic Programming Transcribing English into Logic Fall 2007 Predicate Logic and Quantifiers
- 2. 2/ 33 Further Computer Science & Engineering 235 Examples & Exercises Introduction to Discrete Mathematics Sections 1.3–1.4 of Rosen cse235@cse.unl.edu Introduction Predicate Logic and Quantifiers CSE235 Consider the following statements: x > 3, x = y + 3, x + y = z
- 3. 3/ 33 Propositional Functions Introduction Propositional Functions Propositional Functions Quantifiers Logic Programming Transcribing English into Logic Further Examples & Exercises The truth value of these statements has no meaning without specifying the values of x,y,z. However, we can make propositions out of such statements. A predicate is a property that is affirmed or denied about the subject (in logic, we say “variable” or “argument”) of a statement. “ x is greater than 3” |{z} | {z } subject predicate
- 4. 4/ 33 Terminology: affirmed = holds = is true; denied = does not hold = is not true. |{z} | {z } Propositional subject predicate Functions Predicate Logic and Quantifiers CSE235 Introduction To write in predicate logic: “ x is greater than 3” Propositional Functions Quantifiers Logic Programming Transcribing English into We introduce a (functional) symbol for the predicate, and put the subject as an argument (to the functional symbol): P(x) Examples:
- 5. 5/ 33 Propositional Functions LogicFather(x): unary predicate Further Brother(x,y): binary predicate Examples & Exercises Sum(x,y,z): ternary predicate P(x,y,z,t): n-ary predicate
- 6. Examples & Exercises 6/ 33 Propositional Functions Predicate Logic and Quantifiers Definition CSE235 A statement of the form P(x1,x2,...,xn) is the value of the Introduction propositional function P. Here, (x1,x2,...,xn) is an n-tuple Propositional and P is a predicate. Functions Propositional Universe of Discourse Quantifiers You can think of a propositional function as a function that
- 7. Examples & Exercises 7/ 33 Propositional Functions Functions Evaluates to true or false. Logic Programming Takes one or more arguments. Transcribing intoExpresses a predicate involving the argument(s). English Logic Becomes a proposition when values are assigned to the Further arguments.
- 8. Examples & Exercises 8/ 33 Propositional Functions Example Predicate Logic and Quantifiers Example CSE235 Let Q(x,y,z) denote the statement “x2 + y2 = z2”. What is Introduction the truth value of Q(3,4,5)? What is the truth value of Propositional Functions Q(2,2,3)? How many values of (x,y,z) make the predicate Propositional true?
- 9. Examples & Exercises 9/ 33 Propositional Functions Functions Universe of Discourse Quantifiers Logic Programming Transcribing English into Logic Further Example Predicate Logic and
- 10. Examples & Exercises 10/ 33 Propositional Functions Quantifiers Example Universe of Discourse
- 11. Examples & Exercises 11/ 33 Propositional Functions CSE235 Let Q(x,y,z) denote the statement “x2 + y2 = z2”. What is Introduction the truth value of Q(3,4,5)? What is the truth value of Propositional Quantifiers Logic Programming Transcribing English into Logic Further Since 32 + 42 = 25 = 52, Q(3,4,5) is true. Since 22 + 22 = 8 6= 32 = 9, Q(2,2,3) is false. There are infinitely many values for (x,y,z) that make this propositional function true—how many right triangles are there?
- 12. Examples & Exercises 7/ 33 Propositional Functions Functions Q(2,2,3)? How many values of (x,y,z) make the predicate Propositional true? Functions
- 14. Examples & Exercises 6/ 33 Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Consider the previous example. Does it make sense to assign to x the value “blue”? Intuitively, the universe of discourse is the set of all things we
- 15. Examples & Exercises 7/ 33 Quantifiers Logic Programming sensibly assign to a variable in a propositional function.
- 16. Transcribing English into Logic Further8/ 33 Examples & Transcribing English into Logic What would be the universe of discourse for the propositional function P(x) = “The test will be on x the 23rd” be?
- 17. Examples & Exercises 9/ 33 Universe of Discourse Universe of wish to talk about; that is, the set of all objects that we can Discourse Further
- 18. Transcribing English into Logic Further10/ 33 Examples & Universe of Discourse Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Moreover, each variable in an n-tuple may have a different universe of discourse. Let P(r,g,b,c) = “The rgb-value of the color c is (r,g,b)”.
- 19. Examples & Exercises 11/ 33 Quantifiers Logic Programming For example, P(255,0,0,red) is true, while P(0,0,255,green) is false.
- 20. Transcribing English into Logic Further12/ 33 Examples & Multivariate Functions Universe of Discourse Transcribing English into Logic What are the universes of discourse for (r,g,b,c)?
- 22. Transcribing English into Logic Further14/ 33 Examples & Quantifiers Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions A predicate becomes a proposition when we assign it fixed values. However, another way to make a predicate into a proposition is to quantify it. That is, the predicate is true (or false) for all possible values in the universe of discourse or for
- 23. Examples & Exercises 15/ 33 Introduction Quantifiers Universal Quantifier Existential Quantifier Mixing Quantifiers Binding some value(s) in the universe of discourse. Such quantification can be done with two quantifiers: the universal quantifier and the existential quantifier. Variables Negation
- 24. Transcribing English into Logic Further16/ 33 Examples & Logic Programming
- 25. Transcribing English into Logic Further17/ 33 Examples & Universal Quantifier Definition Predicate
- 26. Transcribing English into Logic Further18/ 33 Examples & Universal Quantifier Example I Logic and Quantifiers CSE235 Introduction Propositional Functions Definition P (x) P (x) x ∀xP (x) which can be read “for all x
- 27. Examples & Universal Quantifier Example I Existential Quantifier
- 28. Transcribing English into Logic Further20/ 33 Examples & Universal Quantifier Example I Propositional Functions Quantifiers Universal Quantifier ∀xP(x) ⇐⇒ P(n1) ∧ P(n2) ∧ ··· ∧ P(nk) Mixing Quantifiers Binding Variables Negation Logic Programming If the universe of discourse is finite, say {n1,n2,...,nk}, then the universal quantifier is simply the conjunction of all elements:
- 29. Examples & Universal Quantifier Example I Predicate Logic and Quantifiers CSE235 Let P(x) be the predicate “x must take a discrete mathematics course” and let Q(x) be the predicate “x is a computer science student”. IntroductionThe universe of discourse for both P(x) and Q(x) is all PropositionalUNL students. Functions Express the statement “Every computer science student Propositional Functions Quantifiers must take a discrete mathematics course”.
- 30. Transcribing English into Logic Further22/ 33 Examples & Universal Quantifier Example I Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Variables Negation Express the statement “Everybody must take a discrete mathematics course or be a computer science student”. Logic Programming
- 31. Examples & Universal Quantifier Example I Predicate Logic and Quantifiers CSE235 Let P(x) be the predicate “x must take a discrete mathematics course” and let Q(x) be the predicate “x is a computer science student”. IntroductionThe universe of discourse for both P(x) and Q(x) is all PropositionalUNL students. Functions Express the statement “Every computer science student
- 32. Transcribing English into Logic Further24/ 33 Examples & Universal Quantifier Example I UniversalQuantifier ∀x(Q(x) → P(x)) Existential Quantifier Mixing Quantifiers Propositional Functions Quantifiers must take a discrete mathematics course”.
- 33. Examples & Universal Quantifier Example I Binding Variables Negation Express the statement “Everybody must take a discrete mathematics course or be a computer science student”. Logic Programming Transcribing English into Logic Further10 /33
- 34. Examples & Universal Quantifier Example I Predicate Logic and Quantifiers CSE235 Let P(x) be the predicate “x must take a discrete mathematics course” and let Q(x) be the predicate “x is a computer science student”. IntroductionThe universe of discourse for both P(x) and Q(x) is all PropositionalUNL students. Functions Express the statement “Every computer science student
- 35. Examples & Universal Quantifier Example I Propositional Functions Quantifiers must take a discrete mathematics course”. Existential Quantifier Mixing Quantifiers
- 36. Examples & Universal Quantifier Example II UniversalQuantifier ∀x(Q(x) → P(x)) Transcribing English into Logic Further10 /33 Predicate Logic and Quantifiers CSE235 Let P(x) be the predicate “x must take a discrete mathematics course” and let Q(x) be the predicate “x is a computer science student”. Binding Variables Negation Logic Programming Express the statement “Everybody must take a discrete mathematics course or be a computer science student”. ∀x(Q(x) ∨ P(x))
- 37. Examples & Universal Quantifier Example I IntroductionThe universe of discourse for both P(x) and Q(x) is all PropositionalUNL students. Functions Propositional Functions Quantifiers must take a discrete mathematics course”. Existential Quantifier Mixing Quantifiers
- 38. Examples & Universal Quantifier Example II Express the statement “Every computer science student UniversalQuantifier ∀x(Q(x) → P(x)) Predicate Logic and Binding Variables Negation Logic Programming Transcribing English into Logic Further10 /33 Express the statement “Everybody must take a discrete mathematics course or be a computer science student”. ∀x(Q(x) ∨ P(x)) Are hetse statements true or false?
- 39. Examples & Universal Quantifier Example I Quantifiers Express the statement “for every x and for every y, x+y > 10” CSE235 Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Variables Negation
- 40. Examples & Universal Quantifier Example II Logic Programming Transcribing English into Logic Further11 /33
- 41. Examples & Universal Quantifier Example II Predicate Logic and Quantifiers Propositional Functions Quantifiers
- 42. Examples & Universal Quantifier Example II Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Variables Negation Logic Programming Transcribing English into Logic CSE235 Introduction Propositional Functions Express the statement “for every x and for every y, x+y > 10” Let P(x,y) be the statement x + y > 10 where the universe of discourse for x,y is the set of integers.
- 43. Transcribing English into Logic Further11 /33 Examples & Universal Quantifier Example II Further11 /33 Predicate Logic and Quantifiers ∀x∀yP(x,y)
- 44. Examples & Universal Quantifier Example II Quantifiers Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Variables Negation Logic CSE235 Introduction Propositional Functions Propositional Functions Express the statement “for every x and for every y, x+y > 10” Let P(x,y) be the statement x + y > 10 where the universe of discourse for x,y is the set of integers. Answer:
- 45. Transcribing English into Logic Further13 /33 Examples & Universal Quantifier Example II Programming Transcribing English into Logic Further11 /33 Predicate Logic and
- 46. Examples & Universal Quantifier Example II Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Quantifiers Express the statement “for every x and for every y, x+y > 10” Let P(x,y) be the statement x + y > 10 where the universe of discourse for x,y is the set of integers. Answer: ∀x∀yP(x,y)
- 47. Transcribing English into Logic Further15 /33 Examples & Universal Quantifier Example II Universal Quantifier ∀x,yP(x,y) Logic Programming Existential Quantifier Mixing Quantifiers Binding Variables Negation Note that we can also use the shorthand
- 48. Transcribing English into Logic Further16 /33 Examples & Existential Quantifier Definition Predicate
- 49. Transcribing English into Logic Further17 /33 Examples & Existential Quantifier Logic and Quantifiers CSE235 Introduction Propositional Functions Definition P (x) x P (x) is true.” We use the notation ∃xP (x) which can be read “there exists anx
- 50. Transcribing English into Logic Further18 /33 Examples & Existential Quantifier Example II Mixing Quantifiers Again, if the universe of discourse is finite, {n1,n2,...,nk}, then
- 51. Transcribing English into Logic Further19 /33 Examples & Existential Quantifier Propositional Functions Quantifiers Universal Quantifier Existential Quantifier ∃xP(x) ⇐⇒ P(n1) ∨ P(n2) ∨ ··· ∨ P(nk) Binding Variables Negation Logic Programming the existential quantifier is simply the disjunction of all elements:
- 52. Transcribing English into Logic Further20 /33 Examples & Existential Quantifier Example II Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Let P(x,y) denote the statement, “x + y = 5”. What does the expression, ∃x∃yP(x,y)
- 53. Transcribing English into Logic Further21 /33 Examples & Existential Quantifier Example I Existential mean? Quantifier Mixing Quantifiers Binding What universe(s) of discourse make it true? Variables Negation Logic Programming
- 54. Transcribing English into Logic Further22 /33 Examples & Existential Quantifier Example II Predicate Logic and Quantifiers Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier CSE235 Introduction Express the statement “there exists a real solution to ax2 + bx − c = 0”
- 55. Transcribing English into Logic Further23 /33 Examples & Existential Quantifier Mixing Quantifiers Binding Variables Negation Logic Programming
- 56. Examples & Existential Quantifier Example II Predicate Logic and Quantifiers Propositional Functions Let P(x) be the statementwhere the universe Universal Quantifier Existential Quantifier Mixing Quantifiers Binding CSE235 Introduction Express the statement “there exists a real solution to ax2 + bx − c = 0” Propositional Functions Quantifiers of discourse for x is the set of reals. Note here that a,b,c are all fixed constants.
- 57. Examples & Existential Quantifier Example II Continued Variables Negation Logic Programming Transcribing English into Logic Further14 /33 Predicate Logic and Quantifiers Propositional CSE235 Introduction Express the statement “there exists a real solution to ax2 + bx − c = 0”
- 58. Examples & Existential Quantifier Example II Functions Let P(x) be the statementwhere the universe The statement can thus be expressed as Existential Quantifier Mixing Propositional Functions Quantifiers Universal Quantifier of discourse for x is the set of reals. Note here that a,b,c are all fixed constants.
- 59. Examples & Existential Quantifier Example II Continued Quantifiers Binding Variables Negation ∃xP(x) Logic Programming Transcribing English into Logic Further14 /33 Predicate Logic and Quantifiers CSE235 Introduction Question: what is the truth value of ∃xP(x)? Propositional Functions
- 60. Examples & Existential Quantifier Example II Propositional Functions Quantifiers Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Variables Negation Logic Programming Transcribing English into Logic Further15 /33
- 62. Examples & Existential Quantifier Example II Continued Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Question: what is the truth value of xP(x)? Answer: it is false. For any real numbers such that b2 < 4ac, there will only be complex solutions, for these cases no such real number x can satisfy the predicate. How can we make it so that it is true?
- 63. Transcribing English into Logic Further15 /33 Examples & Existential Quantifier Example II Continued ∃ Existential Quantifier Mixing Quantifiers Binding Variables Negation
- 64. Transcribing English into Logic Further16 /33 Examples & Logic Programming Transcribing English into Logic Further15 /33 Predicate Logic and Quantifiers CSE235
- 65. Transcribing English into Logic Further17 /33 Examples & Existential Quantifier Example II Continued ∃ Existential Quantifier Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Question: what is the truth value of xP(x)? Answer: it is false. For any real numbers such that b2 < 4ac, there will only be complex solutions, for these cases no such real number x can satisfy the predicate. How can we make it so that it is true?
- 66. Transcribing English into Logic Further18 /33 Examples & Mixing Quantifiers Binding Variables Negation Answer: change the universe of discourse to the complex numbers, C. Logic Programming
- 67. Transcribing English into Logic Further19 /33 Examples & Existential Quantifier Example II Continued Quantifiers Truth Values Functions Predicate Logic and Quantifiers CSE235 Introduction Propositional In general, when are quantified statements true/false? Statement True When False When
- 68. Transcribing English into Logic Further20 /33 Examples & Propositional Functions Quantifiers Universal Quantifier Existential Quantifier Mixing Logic Programming ∀xP(x) P(x) is true for every x. There is an x for which P(x) is false. ∃xP(x) There is an x for which P(x) is true. P(x) is false for every x. Quantifiers Binding Variables Negation Table: Truth Values of Quantifiers
- 69. Transcribing English into Logic Further21 /33 Examples & Mixing Quantifiers Predicate Logic and Quantifiers CSE235
- 70. Transcribing English into Logic Further22 /33 Examples & Mixing Quantifiers I Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier Existential and universal quantifiers can be used together to quantify a predicate statement; for example, ∀x∃yP(x,y) is perfectly valid. However, you must be careful—it must be read left to right.
- 71. Transcribing English into Logic Further23 /33 Examples & Mixing Quantifiers Mixing Quantifiers Logic Programming II Binding Variables Negation For example, ∀x∃yP(x,y) is not equivalent to ∃y∀xP(x,y). Thus, ordering is important.
- 72. Transcribing English into Logic Further24 /33 Examples & Mixing Quantifiers Predicate Logic and Quantifiers CSE235 For example: Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier Those expressions do not mean the same thing! Note that ∃y∀xP(x,y) → ∀x∃yP(x,y), but the converse does not hold
- 73. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Binding However, you can commute similar quantifiers; ∃x∃yP(x,y) is
- 74. Transcribing English into Logic Further26 /33 Examples & Mixing Quantifiers Introduction∀x∃yLoves(x,y): everybody loves somebody Propositional∃y∀xLoves(x,y): There is someone loved by everyone Variables Negation Logic Programming equivalent to ∃y∃xP(x,y) (which is why our shorthand was valid).
- 75. Transcribing English into Logic Further27 /33 Examples & Mixing Quantifiers Mixing Quantifiers Truth Values Predicate Logic and Statement True When False When
- 76. Transcribing English into Logic Further28 /33 Examples & Mixing Quantifiers ∀x∀yP(x,y) P(x,y) is true for every pair x,y. There is at least one pair, x,y for which P(x,y) is false. ∀x∃yP(x,y) For every x, there is a y for which P(x,y) is true. There is an x for which P(x,y) is false for every y.
- 77. Transcribing English into Logic Further29 /33 Examples & Mixing Quantifiers Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier ∃x∀yP(x,y) There is an x for which P(x,y) is true for every y. For every x, there is a y for which P(x,y) is false. ∃x∃yP(x,y) There is at least one pair x,y for which P(x,y) is true. P(x,y) is false for every pair x,y.
- 78. Transcribing English into Logic Further30 /33 Examples & Mixing Quantifiers Mixing Quantifiers Binding Variables Negation Logic Programming Table: Truth Values of 2-variate Quantifiers
- 79. Transcribing English into Logic Further31 /33 Examples & Mixing Quantifiers Example I Predicate Logic and Quantifiers CSE235 Express, in predicate logic, the statement that there are an Introduction infinite number of integers. Propositional Functions Propositional Functions Quantifiers Universal Quantifier
- 80. Examples & Mixing Quantifiers Example I Existential Quantifier Mixing Quantifiers Binding Variables Negation Logic Programming Predicate Logic and Quantifiers CSE235 Quantifiers
- 81. Examples & Mixing Quantifiers Example I Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Variables Negation Logic Programming Introduction Propositional Functions Propositional Functions Express, in predicate logic, the statement that there are an infinite number of integers. Let P(x,y) be the statement that x < y. Let the universe of discourse be the integers, Z.
- 82. Examples & Mixing Quantifiers Example I Transcribing English into Logic Further20 /33 Predicate Logic and Quantifiers
- 83. Examples & Mixing Quantifiers Example I CSE235 Mixing ∀x∃yP(x,y) Quantifiers Binding Variables Negation Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier Express, in predicate logic, the statement that there are an infinite number of integers. Let P(x,y) be the statement that x < y. Let the universe of discourse be the integers, Z. Then the statement can be expressed by the following.
- 84. Examples & Mixing Quantifiers Example I Logic Programming Transcribing English into Logic Further20 /33
- 85. Transcribing English into Logic Further21 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Predicate Logic and Quantifiers CSE235 Express the commutative law of addition for R. Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier
- 86. Transcribing English into Logic Further21 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Existential Quantifier Mixing Quantifiers Binding Variables Negation Logic Programming Predicate Logic and
- 87. Transcribing English into Logic Further21 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Express the commutative law of addition for R. We want to express that for every pair of reals, x,y the following identity holds: x + y = y + x
- 88. Transcribing English into Logic Further21 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Quantifiers Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Variables Negation Logic Programming Predicate Logic and
- 89. Transcribing English into Logic Further21 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier Express the commutative law of addition for R. We want to express that for every pair of reals, x,y the following identity holds: x + y = y + x Then we have the following:
- 90. Transcribing English into Logic Further21 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Mixing QuantifiersBinding ∀x∀y(x + y = y + x) Variables Negation Logic Programming
- 91. Transcribing English into Logic Further22 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Continued Predicate Logic and Quantifiers CSE235 Express the multiplicative inverse law for (nonzero) rationals Introduction Q {0}. Propositional Functions Propositional Functions Quantifiers
- 92. Transcribing English into Logic Further22 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Continued Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Variables Negation Logic Programming Predicate Logic and Quantifiers
- 93. Transcribing English into Logic Further22 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Continued CSE235 Quantifiers Universal Quantifier Existential Quantifier Mixing Introduction Propositional Functions Propositional Functions Express the multiplicative inverse law for (nonzero) rationals Q {0}. We want to express that for every real number x, there exists a real number y such that xy = 1.
- 94. Transcribing English into Logic Further22 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Continued Quantifiers Binding Variables Negation Logic Programming Predicate Logic and Quantifiers
- 95. Transcribing English into Logic Further22 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Continued CSE235 Mixing ∀x∃y(xy = 1) Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier Express the multiplicative inverse law for (nonzero) rationals Q {0}. We want to express that for every real number x, there exists a real number y such that xy = 1. Then we have the following:
- 96. Transcribing English into Logic Further22 /33 Examples & Mixing Quantifiers Example II: More Mathematical Statements Continued Quantifiers Binding Variables Negation Logic Programming
- 97. Transcribing English into Logic Further23 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements Predicate Logic and Quantifiers CSE235 Introduction Is commutativity for subtraction valid over the reals? Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier
- 98. Transcribing English into Logic Further23 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements Mixing Quantifiers Binding Variables Negation Logic Programming Predicate Logic and Quantifiers
- 99. Transcribing English into Logic Further23 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements CSE235 Quantifiers Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Introduction Propositional Functions Propositional Functions Is commutativity for subtraction valid over the reals? That is, for all pairs of real numbers x,y does the identity x − y = y − x hold? Express this using quantifiers.
- 100. Transcribing English into Logic Further23 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements Variables Negation Logic Programming Predicate Logic and Quantifiers
- 101. Transcribing English into Logic Further23 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements CSE235 Mixing Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier Is commutativity for subtraction valid over the reals? That is, for all pairs of real numbers x,y does the identity x − y = y − x hold? Express this using quantifiers. The expression is ∀x∀y(x − y = y − x)
- 102. Transcribing English into Logic Further23 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements Quantifiers Binding Variables Negation Logic Programming
- 103. Transcribing English into Logic Further24 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements Continued Predicate Logic and Quantifiers CSE235 Introduction Is there a multiplicative inverse law over the nonzero integers? Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier
- 104. Transcribing English into Logic Further24 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements Continued Mixing Quantifiers Binding Variables Negation Logic Programming Predicate Logic and Quantifiers CSE235
- 105. Transcribing English into Logic Further24 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements Continued Functions Quantifiers Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Introduction Propositional Functions Propositional Is there a multiplicative inverse law over the nonzero integers? That is, for every integer x does there exists a y such that xy = 1?
- 106. Transcribing English into Logic Further24 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements Continued Variables Negation Logic Programming Predicate Logic and Quantifiers
- 107. Transcribing English into Logic Further24 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements Continued CSE235 Mixing statement held, then 5 = 1/y, but for any (nonzero) integer y, Introduction Propositional Functions Propositional Functions Quantifiers Universal Quantifier Existential Quantifier Is there a multiplicative inverse law over the nonzero integers? That is, for every integer x does there exists a y such that xy = 1? This is false, since we can find a counter example. Take any integer, say 5 and multiply it with another integer, y. If the
- 108. Transcribing English into Logic Further24 /33 Examples & Mixing Quantifiers Example II: False Mathematical Statements Continued Quantifiers Binding Variables |1/y| ≤ 1. Negation Logic Programming
- 109. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise Logic andPredicate Express the statement “there is a number x such that Quantifiers when it is added to any number, the result is that CSE235 number, and if it is multiplied by any number, the Introduction result is x” as a logical expression. Propositional Functions Propositional Solution: Functions
- 110. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise Quantifiers Universal Quantifier Existential Quantifier Mixing Quantifiers Binding Variables Negation Logic Programming Universal Quantifier Existential Quantifier Mixing
- 111. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise Quantifiers Binding Variables Negation Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Quantifiers Express the statement “there is a number x such that when it is added to any number, the result is that number, and if it is multiplied by any number, the result is x” as a logical expression. Solution: Let P(x,y) be the expression “x + y = y”.
- 112. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise Logic Programming QuantifiersLet P(x,y) be the expression “x + y = y”. UniversalQuantifierLet Q(x,y) be the expression “xy = x”. Existential Quantifier Mixing
- 113. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise Quantifiers Binding Variables Negation Logic Programming Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Express the statement “there is a number x such that when it is added to any number, the result is that number, and if it is multiplied by any number, the result is x” as a logical expression. Solution:
- 114. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise QuantifiersLet P(x,y) be the expression “x + y = y”. UniversalQuantifierLet Q(x,y) be the expression “xy = x”. ExistentialQuantifierThen the expression is Mixing
- 115. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise Quantifiers Binding Variables ∃x∀y (P(x,y) ∧ Q(x,y)) Negation Logic Programming Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Express the statement “there is a number x such that when it is added to any number, the result is that number, and if it is multiplied by any number, the result is x” as a logical expression. Solution:
- 116. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise QuantifiersLet P(x,y) be the expression “x + y = y”.
- 117. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Express the statement “there is a number x such that when it is added to any number, the result is that number, and if it is multiplied by any number, the result is x” as a logical expression. Solution:
- 118. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise Binding ∃x∀y (P(x,y) ∧ Q(x,y))
- 119. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise UniversalQuantifierLet Q(x,y) be the expression “xy = x”. ExistentialQuantifierThen the expression is Mixing Quantifiers Variables Negation Logic Programming Over what universe(s) of discourse does this statement hold?
- 120. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise QuantifiersLet P(x,y) be the expression “x + y = y”. UniversalQuantifierLet Q(x,y) be the expression “xy = x”. ExistentialQuantifierThen the expression is Mixing
- 121. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise Quantifiers Binding Variables Negation ∃x∀y (P(x,y) ∧ Q(x,y)) Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Express the statement “there is a number x such that when it is added to any number, the result is that number, and if it is multiplied by any number, the result is x” as a logical expression. Solution:
- 122. Transcribing English into Logic Further25 /33 Examples & Mixing Quantifiers Exercise LogicOver what universe(s) of discourse does this statement Programming hold? This is the additive identity law and holds for N,Z,R,Q but does not hold for Z+.
- 123. Examples & Binding Variables I Propositional Functions Example Quantifiers In the expression ∃x∀yP(x,y) both x and y are bound. Universal Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions When a quantifier is used on a variable x, we say that x is bound. If no quantifier is used on a variable in a predicate statement, it is called free.
- 124. Examples & QuantifierExistential In the expression ∀xP(x,y), x is bound, but y is free. Quantifier Mixing Quantifiers BindingVariables A statement is called a well-formed formula, when all variables Negation are properly quantified. Logic Programming Transcribing English into Logic Further26 /33
- 125. Examples & Binding Variables II Propositional Example Functions Quantifiers In the expression ∃x,y∀zP(x,y,z,c) the scope of the Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions The set of all variables bound by a common quantifier is the scope of that quantifier.
- 126. Examples & UniversalQuantifier existential quantifier is {x,y}, the scope of the universal ExistentialQuantifier quantifier is just z and c has no scope since it is free. Mixing Quantifiers Binding Variables Negation Logic Programming Transcribing English into Logic Further27 /33
- 127. Examples & Predicate Logic and Quantifiers CSE235 Introduction Just as we can use negation with propositions, we can use them with quantified expressions.
- 128. Examples & Logic fact if the universe of discourse is finite, it is exactly De
- 130. Examples & Negation Lemma P (x) be a predicate. Then the following hold. ¬∀xP (x) ≡∃ x¬P (x) ¬∃xP (x) ≡∀ x¬P (x)
- 131. Examples & This is essentially a quantified version of De Morgan’s Law (in Further28 /33 Negation Truth Values Predicate Logic and Quantifiers CSE235 Introduction Statement True When False When
- 132. Examples & Propositional Functions Propositional Functions Quantifiers Universal Quantifier Negation Logic Programming Transcribing English into Logic Further29 /33 ¬∃xP(x) ≡ ∀x¬P(x) For every x, P(x) is false. There is an x for which P(x) is true. ¬∀xP(x) ≡ ∃x¬P(x) There is an x for which P(x) is false. P(x) is true for every x. Existential Quantifier Mixing Quantifiers Binding Variables Table: Truth Values of Negated Quantifiers
- 133. Prolog FunctionsProlog allows the user to express facts and rules PropositionalFacts are proposational functions: student(juana), Functions enrolled(juana,cse235), instructor(patel,cse235), etc. Quantifiers Predicate Logic and Quantifiers CSE235 Introduction Propositional Prolog (Programming in Logic) is a programming language based on (a restricted form of) Predicate Calculus. It was developped by the logicians of the artificial intelligence community for symbolic reasoning. Logic Further Examples & Exercises ?enrolled(juana,cse478) ?enrolled(X,cse478) ?teaches(X,juana)
- 134. Rules are implications with conjunctions: Logic Programming teaches(X,Y) :- instructor(X,Z), enrolled(Y,Z) Transcribing English intoProlog answers queries such as: 30 /33 by binding variables and doing theorem proving (i.e., applying inference rules) as we will see in Section 1.5.
- 135. 32 /33 Conclusion Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Quantifiers Logic Programming Logic is more precise than English. Transcribing English to Logic and vice versa can be tricky. When writing statements with quantifiers, usually the correct meaning is conveyed with the following combinations: Use ∀ with ⇒ Example: ∀xLion(x) ⇒ Fierce(x) ∀xLion(x) ∧ Fierce(x) means “everyone is a lion and
- 136. Further Example: ∃xLion(x) ∧ Drinks(x,coffee): holds when you
- 137. 32 /33 Conclusion English into Logic everyone is fierce” Transcribing English into Logic Use ∃ with ∧ Examples & Exercises have at least one lion that drinks coffee
- 138. ∃xLion(x) ⇒ Drinks(x,coffee) holds when you have people even though no lion drinks coffee. 31 /33
- 140. Propositional Logic Exercises Predicate Logic and Quantifiers CSE235 Introduction Examples? Exercises? Rewrite the expression, Functions Propositional Functions Quantifiers Logic Programming Transcribing English into Let P(x,y) denote “x is a factor of y” where x ∈ {1,2,3,...} and y ∈ {2,3,4,...}. Let Q(y) denote
- 141. 32 /33 Conclusion “”. When is Q(y) true?
- 142. 32 /33 Conclusion Predicate Logic and Quantifiers CSE235 Examples? Exercises? Rewrite the expression, Introduction Propositional Functions Answer: Use the negated quantifiers and De Morgan’s law. Propositional Functions Further Examples & Exercises
- 143. 32 /33 Conclusion Quantifiers Logic “”. When is Q(y) true? Predicate Logic and Quantifiers CSE235 Examples? Exercises? Rewrite the expression, Introduction Propositional Functions Answer: Use the negated quantifiers and De Morgan’s law. Logic Programming Transcribing English into Let P(x,y) denote “x is a factor of y” where x ∈ {1,2,3,...} and y ∈ {2,3,4,...}. Let Q(y) denote
- 144. Propositional Functions Quantifiers Logic “”. When is Q(y) true? Answer: Only when y is a prime number. Further Examples & Exercises Logic Programming Transcribing English into Let P(x,y) denote “x is a factor of y” where x ∈ {1,2,3,...} and y ∈ {2,3,4,...}. Let Q(y) denote
- 145. 32 /33 Conclusion Predicate Logic and Quantifiers CSE235 Introduction Propositional Functions Propositional Functions Quantifiers Logic Programming Transcribing English into Logic Some students wondered if ∀x,yP(x,y) ≡ ∀xP(x,y) ∧ ∀yP(x,y) This is certainly not true. In the left-hand side, both x and y are bound. In the right-hand side, x is bound in the first predicate, but y is free. In the second predicate, y is bound but x is free. All variables that occur in a propositional function must be bound to turn it into a proposition.
- 146. Extra Question Further Examples & Exercises Thus, the left-hand side is a proposition, but the right-hand side is not. How can they be equivalent? 33 /33