PREDICATE LOGIC. Schaum's outline chapter 4 Rosen chapter 1. September 11, ioc.pdf

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1 PREDICATE LOGIC Schaum's outline chapter 4 Rosen chapter 1 September 11, 2018 margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

2 Contents 1 Predicates and quantiers 2 Logical equivalences for quantiers 3 Validity and satisability 4 Examples margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

3 Next section 1 Predicates and quantiers 2 Logical equivalences for quantiers 3 Validity and satisability 4 Examples margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

4 Motivation Propositional logic cannot adequately express the meaning of all statements in mathematics and in natural language. Example Having assumptions: Every computer connected to the school network is functioning properly. ELVIS is one of the computers connected to the school network. No rules of propositional logic allow us to conclude the truth of the statement ELVIS is functioning properly Classical example From the propositions All men are mortal. Socrates is a man. we cannot derive that Socrates is mortal. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

5 Motivation Propositional logic cannot adequately express the meaning of all statements in mathematics and in natural language. Example Having assumptions: Every computer connected to the school network is functioning properly. ELVIS is one of the computers connected to the school network. No rules of propositional logic allow us to conclude the truth of the statement ELVIS is functioning properly Classical example From the propositions All men are mortal. Socrates is a man. we cannot derive that Socrates is mortal. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

6 Predicate logic Extends propositional logic by Individuals a,b,..., ELVIS,... (Individual) variables x, y,... Predicates (= propositional functions) P(x), Q(x), R(x, y),... Quantiers, A propositional function is a generalization of proposition: its argument stands for en element from its domain; its value is T or F depending on the property of its argument(s). margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

7 Predicate logic Extends propositional logic by Individuals a,b,..., ELVIS,... (Individual) variables x, y,... Predicates (= propositional functions) P(x), Q(x), R(x, y),... Quantiers, A propositional function is a generalization of proposition: its argument stands for en element from its domain; its value is T or F depending on the property of its argument(s). margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

8 Propositional function Formally, for a given n {0,1,...}, a predicate is a n-ary function (=a function with n arguments) of type P : D 1... D n {T,F } U = D 1... D n is the domain (or the universe) of the predicate P; the statement P(x 1,...,x n ), becomes a proposition (representing a property of its arguments) if either x 1 D 1,...,x n D n or all variables x 1,...,x n are bound by quantiers (see next slides). Example 1 Let P(x) denote x > 2 and U = Z. Then P(5) is true; P(2) is false. Example 2 Let R(x,y,z) denote the statement x + y = z and U = Z. Then R(1,2,3) is true; R(0,0, 5) is false. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

9 Propositional function Formally, for a given n {0,1,...}, a predicate is a n-ary function (=a function with n arguments) of type P : D 1... D n {T,F } U = D 1... D n is the domain (or the universe) of the predicate P; the statement P(x 1,...,x n ), becomes a proposition (representing a property of its arguments) if either x 1 D 1,...,x n D n or all variables x 1,...,x n are bound by quantiers (see next slides). Example 1 Let P(x) denote x > 2 and U = Z. Then P(5) is true; P(2) is false. Example 2 Let R(x,y,z) denote the statement x + y = z and U = Z. Then R(1,2,3) is true; R(0,0, 5) is false. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

10 Propositional function Formally, for a given n {0,1,...}, a predicate is a n-ary function (=a function with n arguments) of type P : D 1... D n {T,F } U = D 1... D n is the domain (or the universe) of the predicate P; the statement P(x 1,...,x n ), becomes a proposition (representing a property of its arguments) if either x 1 D 1,...,x n D n or all variables x 1,...,x n are bound by quantiers (see next slides). Example 1 Let P(x) denote x > 2 and U = Z. Then P(5) is true; P(2) is false. Example 2 Let R(x,y,z) denote the statement x + y = z and U = Z. Then R(1,2,3) is true; R(0,0, 5) is false. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

11 Universal quantier Let U is a given domain of discourse (= universe of discourse or simply domain). Denition The universal quantication of P(x) is the statement P(x) for all values of x in the domain U. The notation xp(x) denotes the universal quantication of P(x). Here is called the universal quantier. We read xp(x) as for all x P(x) or for every x P(x). An element for which P(x) is false is called a counterexample of xp(x). margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

12 Universal quantier Let U is a given domain of discourse (= universe of discourse or simply domain). Denition The universal quantication of P(x) is the statement P(x) for all values of x in the domain U. The notation xp(x) denotes the universal quantication of P(x). Here is called the universal quantier. We read xp(x) as for all x P(x) or for every x P(x). An element for which P(x) is false is called a counterexample of xp(x). Examples Let P(x) be the statement x + 1 > x and domain is Z. Then the quantication xp(x) is true. Let Q(x) be the statement x < 2 and domain is Z. Then the quantication xq(x) is false. Let R(x) be the statement x 2 > 0 and domain is R. Then the quantication xr(x) is false. A counterexample here is x = 0. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

13 Universal quantier Let U is a given domain of discourse (= universe of discourse or simply domain). Denition The universal quantication of P(x) is the statement P(x) for all values of x in the domain U. The notation xp(x) denotes the universal quantication of P(x). Here is called the universal quantier. We read xp(x) as for all x P(x) or for every x P(x). An element for which P(x) is false is called a counterexample of xp(x). Aanother example Let P(x) be the statement x 2 x and domain is R. Then the quantication xp(x) is false. A counterexample is x = 1 2. However, if domain is Z, then the quantication xp(x) is true. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

14 Existential quantier Denition The existential quantication of P(x) is the statement There exists an element x in the domain U such that P(x). We use the notation notation xp(x) for the existential quantication of P(x). Here is called the existential quantier. Remark: The meaning of xp(x) changes when the domain changes. Without specifying the domain, the statement xp(x) has no meaning. Examples Let P(x) be the statement x > 3 and domain is R. Then the quantication xp(x) is true. Let Q(x) be the statement x = x + 1 and domain is R. Then the quantication xq(x) is false. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

15 Existential quantier Denition The existential quantication of P(x) is the statement There exists an element x in the domain U such that P(x). We use the notation notation xp(x) for the existential quantication of P(x). Here is called the existential quantier. Remark: The meaning of xp(x) changes when the domain changes. Without specifying the domain, the statement xp(x) has no meaning. Examples Let P(x) be the statement x > 3 and domain is R. Then the quantication xp(x) is true. Let Q(x) be the statement x = x + 1 and domain is R. Then the quantication xq(x) is false. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

16 Bound and free variables The quantiers are said to bind the variable x in the expressions xp(x) and xp(x). Variables in the scope of some quantier are called bound variables. All other variables in the expression are called free variables. A propositional function that does not contain any free variables is a proposition and has a truth value. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

17 Bound and free variables The quantiers are said to bind the variable x in the expressions xp(x) and xp(x). Variables in the scope of some quantier are called bound variables. All other variables in the expression are called free variables. A propositional function that does not contain any free variables is a proposition and has a truth value. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

18 Bound and free variables The quantiers are said to bind the variable x in the expressions xp(x) and xp(x). Variables in the scope of some quantier are called bound variables. All other variables in the expression are called free variables. A propositional function that does not contain any free variables is a proposition and has a truth value. Examples In the statement x(x + y = 1), the variable x is bound, but the variable y is free; In the statement x(p(x) Q(x)) xr(x), all variables are bound margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

19 Quantiers as conjunctions and/or disjunctions If the domain is nite then universal/existential quantiers can be expressed by conjunctions/disjunctions. If the domain U = {1,2,3,4}, then xp(x) = P(1) P(2) P(3) P(4), and xp(x) = P(1) P(2) P(3) P(4). Even if the domains are innite, you can still think of the quantiers in this fashion, but the equivalent expressions without quantiers will be innitely long. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

20 Quantiers as conjunctions and/or disjunctions If the domain is nite then universal/existential quantiers can be expressed by conjunctions/disjunctions. If the domain U = {1,2,3,4}, then xp(x) = P(1) P(2) P(3) P(4), and xp(x) = P(1) P(2) P(3) P(4). Even if the domains are innite, you can still think of the quantiers in this fashion, but the equivalent expressions without quantiers will be innitely long. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

21 Quantiers as conjunctions and/or disjunctions If the domain is nite then universal/existential quantiers can be expressed by conjunctions/disjunctions. If the domain U = {1,2,3,4}, then xp(x) = P(1) P(2) P(3) P(4), and xp(x) = P(1) P(2) P(3) P(4). Even if the domains are innite, you can still think of the quantiers in this fashion, but the equivalent expressions without quantiers will be innitely long. Example Let P(x) be the statement x 2 < 10. Then it is true for the domain U 1 = {1,2,3}, but it is false for the domain U 2 = {1,2,3,4}. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

22 Examples of nested quantiers Statements with complex semantics require nested quantiers. Example 1 Statement: Every real number has an inverse w.r.t. addition Domain U = R. The property is expressed by x y(x + y = 0) margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

23 Examples of nested quantiers Statements with complex semantics require nested quantiers. Example 1 Statement: Every real number has an inverse w.r.t. addition Domain U = R. The property is expressed by x y(x + y = 0) Example 2 Statement: Every real number except zero has a multiplicative inverse. Domain U = R. The property is expressed by x(x 0 y(xy = 1)) margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

24 Notation: quantiers with restricted domains An abbreviated notation is often used to restrict the domain of a quantier. x < 0.(x 2 > 0) is another way of expressing x(x < 0 x 2 > 0). x A.(0 < x 5) is another way of expressing x(x A (0 < x) (x 5)). z > 0.(z 2 = 2) is another way of expressing z(z > 0 z 2 = 2). margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

25 Precedence of quantiers The quantiers and have higher precedence than all logical operators from propositional calculus For example xp(x) Q(x) means ( xp(x)) Q(x) rather than x(p(x) Q(x)). margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

26 Next section 1 Predicates and quantiers 2 Logical equivalences for quantiers 3 Validity and satisability 4 Examples margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

27 Equivalences Statements involving predicates and quantiers are logically equivalent if and only if they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for the variables in these propositional functions. We use the notation S T to indicate that two statements S and T involving predicates and quantiers are logically equivalent. Examples x. S(x) x.s(x) x.(p(x) Q(x)) x.p(x) x.q(x) x. y.p(x, y) y. x.p(x, y) x. y.p(x, y) y. x.p(x, y) margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

28 Equivalences Statements involving predicates and quantiers are logically equivalent if and only if they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for the variables in these propositional functions. We use the notation S T to indicate that two statements S and T involving predicates and quantiers are logically equivalent. Examples x. S(x) x.s(x) x.(p(x) Q(x)) x.p(x) x.q(x) x. y.p(x, y) y. x.p(x, y) x. y.p(x, y) y. x.p(x, y) margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

29 Equivalences Statements involving predicates and quantiers are logically equivalent if and only if they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for the variables in these propositional functions. We use the notation S T to indicate that two statements S and T involving predicates and quantiers are logically equivalent. Examples x. S(x) x.s(x) x.(p(x) Q(x)) x.p(x) x.q(x) x. y.p(x, y) y. x.p(x, y) x. y.p(x, y) y. x.p(x, y) margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

30 Equivalences Statements involving predicates and quantiers are logically equivalent if and only if they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for the variables in these propositional functions. We use the notation S T to indicate that two statements S and T involving predicates and quantiers are logically equivalent. Examples x. S(x) x.s(x) x.(p(x) Q(x)) x.p(x) x.q(x) x. y.p(x, y) y. x.p(x, y) x. y.p(x, y) y. x.p(x, y) margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

31 Equivalences Statements involving predicates and quantiers are logically equivalent if and only if they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for the variables in these propositional functions. We use the notation S T to indicate that two statements S and T involving predicates and quantiers are logically equivalent. Examples x. S(x) x.s(x) x.(p(x) Q(x)) x.p(x) x.q(x) x. y.p(x, y) y. x.p(x, y) x. y.p(x, y) y. x.p(x, y) margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

32 Quantications of two variables. Example where x. y.p(x,y) y. x.p(x,y) Let P(x,y) denotex + y = 0, where the domain for all variables consists of all real numbers. y. x.p(x, y) means the proposition There is a real number y such that for every real number x, x + y = 0. Actually, no matter what value of y is chosen, there is only one value of x for which x + y = 0. Hence, the proposition y. x.p(x,y) false. x. y.p(x, y) means the proposition For every real number x there is a real number y such that x + y = 0. In fact, given a real number x, there is a real number y such that x + y = 0; namely, y = x. Hence, the statement is true. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

33 Quantications of two variables. The meanings of the dierent possible quantications involving two variables. ICY0001: Lecture 2 September 11, / 25

34 De Morgan's law for quantiers The rules for negating quantiers are: ( x.p(x)) x. P(x) ( x.p(x)) x. P(x) margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

35 Next section 1 Predicates and quantiers 2 Logical equivalences for quantiers 3 Validity and satisability 4 Examples margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

36 Validity An assertion in predicate calculus is logically valid (or simply valid) if it is true in every interpretation, that is i it is true for all domains for every propositional functions substituted for the predicates in the assertion Valid assertions in predicate logic play a role similar to tautologies in propositional logic. Example x.(p(x) P(x)) margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

37 Validity An assertion in predicate calculus is logically valid (or simply valid) if it is true in every interpretation, that is i it is true for all domains for every propositional functions substituted for the predicates in the assertion Valid assertions in predicate logic play a role similar to tautologies in propositional logic. Example x.(p(x) P(x)) margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

38 Satisability An assertion in predicate calculus is satisable i it is true for some domain for some propositional functions that can be substituted for the predicates in the assertion Valid assertions in predicate logic play a role similar to tautologies in propositional logic. Examples x. y.p(x, y) is satisable The domain N, and the propositional function satisfy this assertion. x.(p(x) P(x)) is unsatisable margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

39 Satisability An assertion in predicate calculus is satisable i it is true for some domain for some propositional functions that can be substituted for the predicates in the assertion Valid assertions in predicate logic play a role similar to tautologies in propositional logic. Examples x. y.p(x, y) is satisable The domain N, and the propositional function satisfy this assertion. x.(p(x) P(x)) is unsatisable margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

40 Next section 1 Predicates and quantiers 2 Logical equivalences for quantiers 3 Validity and satisability 4 Examples margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

41 Example 1: If x is irrational then so is x Let Irrational(x) denote the propositional function x is irrational, and Rational(x) = Irrational(x) Proposition x R +. Irrational(x) Irrational( x) Proof. Let x be positive real number. We will show the contrapositive, i.e. x R +. Rational( x) Rational(x) In other words we prove that if x is rational then so is x. Assume that x is a rational number. Then, by denition, there must exists two natural numbers m and n such that x = m/n. But then x = m 2 /n 2 and, since m 2 and n 2 are natural numbers, which by denition implies that x is a rational number as required. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

42 Example 2: n is even i n 2 is even Let Even(x) denote the propositional function x is even, and Odd(x) = Even(x) Proposition n Z. Even(n) Even(n 2 ) Proof. Let n Z Necessity. Let's assume that n is even, i.e. there exists an integer k such that n = 2k. Then n 2 = (2k) 2 = 4k 2 = 2(2k 2 ), and thus for l = 2k 2,n 2 = 2l and so is even. Suciency. We will show the contrapositive,i.e. Odd(n) Odd(n 2 ). Let's assume that n is odd, i.e. there exists an integer k such that n = 2k + 1. Then n 2 = (2k + 1) 2 = 2(2k 2 + 2k) + 1,and thus for l = 2k 2 + k,n 2 = 2l + 1 and so is odd. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

43 Example 3: 4k = i 2 j 2 Proposition k N. i N. j N. 4k = i 2 j 2 Proof. Let k N. Let i = k + 1 and j = k 1. i 2 j 2 = (k + 1) 2 (k 1) 2 = k 2 + 2k + 1 k 2 + 2k 1 = 4k. margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

44 Example 4: Either n 2 0 (mod 4) or n 2 1 (mod 4) Remark: Formula a b (mod 4) stands for the proposition b is the remainder after division of a by 4. Proposition n Z. n 2 0 (mod 4) n 2 1 (mod 4) Proof. Let n Z. n is either even or odd. We consider each case separately. (1). Assume n is even. Then there exists m such that n = 2m. But then n 2 = 4m 2 0 (mod 4). (2). Assume n is odd. Then there exists m such that n = 2m + 1. But then n 2 = 4m 2 + 4m + 1 = 4(m 2 + m) (mod 4). margarita.spitsakova@ttu.ee ICY0001: Lecture 2 September 11, / 25

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