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Discrete Mathematics -- Chapter Ch t 2: 2 F Fundamentals d t l off Logic Hung-Yu Kao (高宏宇) Department of Computer Science and Information Engineering, N National l Cheng Ch Kung K University U Outline z z z z z Basic Connectives and Truth Tables Logical Equivalence: The Law of Logic Logical Implication: Rule of Inference The Use of Quantifiers Quantifiers Definitions Quantifiers, Definitions, and the Proofs of Theorems 2009 Spring Discrete Mathematics – CH2 2 The Way to Proof 證明”A君是萬能”這句話是錯的 z (1) (2) (3) A君可以搬動任何石頭 A君可以製造出他無法搬動的石頭 (1) (2) 相互矛盾, (1), 相互矛盾 原命題為假 A logical sequence of statements. 2009 Spring Discrete Mathematics – CH2 3 2.1 Basic Connectives and Truth Tables z z Statement 敘述 (Proposition S (P i i 命題): 命題) are declarative d l i sentences that are either true or false, but not both. Primitive Statement (原始命題) z Examples z p: ‘Discrete Discrete Mathematics Mathematics’ is a required course for sophomores. z q: Margaret Mitchell wrote ‘Gone with the Wind’. z r: 2+3=5. z “What a beautiful evening!” (not a statement) z ”Get up and do your exercises.” (not a statement) z No way to make them simpler “The number x is an integer.” g is a statement ? 2009 Spring Discrete Mathematics – CH2 4 2.1 Basic Connectives and Truth Tables z New statements can be obtained from primitive statements in two ways z z 2009 Spring Transform a given statement p into the statement ¬p, which denotes its negation and is read “Not Not p p”. 非p (Negation statements) Combine two or more statements into a compound statement, using logical connectives. (Compound statements複合敘述) Discrete Mathematics – CH2 5 2.1 Basic Connectives and Truth Tables z Compound Statement (Logical Connectives) z z z z z Conjunction: j p רq (read “p and q”) Disjunction : p v q (read “p or q”) Exclusive or : p v q. Implication: p → q, p: hypothesis “p p implies q q”. q: conclusion l i Biconditional: p ↔ q z z z 2009 Spring “p p if and only if q q” (若且為若) “p iff q” “p p is necessary and sufficient for q q”. Discrete Mathematics – CH2 p → q is also called called, • If p, then q • p is sufficient for q • p is a sufficient condition for q • q is necessary for p • q is a necessary condition for p • p only if q •q q whenever p 6 2.1 Basic Connectives and Truth Tables z The truth and falsity of the compound statements based on the truth values of their components (primitive statements). We do not want a true statement to lead us into believing something that is false. p ¬p 0 1 1 0 p q p רq pvq pvq p→q p↔q 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 0 0 1 Truth Tables 2009 Spring Discrete Mathematics – CH2 7 Writing Down Truth Tables z “True, False” is preferred (less ambiguous) than “1, 0” z List the elementary truth values in a consistent way (from F…F to T…T or from T…T to F…F). z They get long: with n variables, the length is 2n. z In the end we want to be able to reason about logic without having to write down such tables. 2009 Spring Discrete Mathematics – CH2 8 2.1 Basic Connectives and Truth Tables z z z p→q is equivalent with (¬p ∨ q) It’s not relative about the causal relationship z If Discrete Mathematics Mathematics’ is a required course for sophomores, then Margaret Mitchell wrote ‘Gone with the Wind’. is true z If “2+3=5”, then “4+2=6” is true z If “2+3=6”, 2 3 6 , then “2+4=7” 2 4 7 is true “Margaret Mitchell wrote ‘Gone with the Wind Wind’(q), and 2+3 ≠ 5 (not r), the ‘Discrete Mathematics’ is a required course for sophomores (p). z 2009 Spring 1 2 3 q ∧ ( ¬r → p ) Discrete Mathematics – CH2 9 Implications examples p: If it is sunny today, today then we will go to the beach. beach q: If today is Friday, Friday then 2+3=5. 2+3=5 r: If today is Friday, then 2+3=6. r is true every day except Friday, even though 2+3=6 is false. The M Th Mathematical h i l concept off an implication i li i is i independent i d d off a cause-and-effect relationship between hypothesis and conclusion. 2009 Spring Discrete Mathematics – CH2 10 2.1 Basic Connectives and Truth Tables z Ex 2.1: Let s, t, and u denote the primitive statements. z z z z s: Phyllis goes out for a walk. t: The moon is out. out u: It is snowing. English sentences for compound statements. z Same? z z 2009 Spring (t ¬ רu) → s: z If the th moon is i outt and d it is i nott snowing, i then th Phyllis Ph lli goes outt for f a walk. lk t → (¬u → s) : z If the moon is out, then if it is not snowing Phyllis goes out for a walk. ¬ (s ↔ (u v t)) : z It is not the case that Phyllis goes out for a walk if and only if it is snowingg or the moon is out. Discrete Mathematics – CH2 11 2.1 Basic Connectives and Truth Tables z Let s, t, and u denote the primitive statements. z z z z s: Phyllis goes out for a walk. t: The moon is out. out u: It is snowing. Reversely, R l examine i the th logical l i l form f for f given i English E li h sentences. t z “Phyllis will go out walking if and only if the moon is out.” z z t (u ¬ רtt ) → ¬ s It is snowing but Phyllis will still go out for a walk. z 2009 Spring ↔ If it is snowing and the moon is not out, then Phyllis will not go out for a walk.” z z s uרs Discrete Mathematics – CH2 12 2.1 Basic Connectives and Truth Tables z Ex 2.3: z Decision ((selection)) structure z z 2009 Spring In computer science, the if-then and if-then-else decision structure arise in high-level programming languages such as Java and C++. E.g., “if p then q else r,” q is executed when p is true and r i executedd when is h p is i false. f l Discrete Mathematics – CH2 13 2.1 Basic Connectives and Truth Tables z Tautology, T0: If a compound statement is true for all truth value assignments for its component statements. z z Contradiction, F0: If a compound statement is false for all truth value assignments for its component statements. z z Example: p∨¬p Example: p∧¬p Examples z z 2009 Spring “2 = 3–1” is not a tautology, but “2=1 or 2≠1” is; “1+1=3” is not a contradiction, but “1=1 and 1≠1” is. Discrete Mathematics – CH2 14 2.1 Basic Connectives and Truth Tables z An argument starts with a list of given statements called premises (hypothesis) and a statement called the conclusion of the argument. z 2009 Spring (p1 רp2ר … רpn) → q Discrete Mathematics – CH2 15 2.2 Logical Equivalence: The Laws of Logic z Logically equivalent,s1 ⇔ s2 : When the statement s1 is true (false) if and only if s2 is true (false). ¬p ∨ q ⇔ p → q ( p → q) ∧ (q → p) ⇔ p ↔ q the same truth tables 2009 Spring p q ¬p ¬pvq 0 0 1 1 0 1 0 1 1 1 0 0 1 1 0 1 p→q q→p 1 1 0 1 ( → q ) ( רq → p)) (p p↔q 1 0 0 1 1 0 0 1 1 0 1 1 Discrete Mathematics – CH2 16 2.2 Logical Equivalence: The Laws of Logic z Negation, v (exclusive) 2009 Spring Discrete Mathematics – CH2 17 2.2 Logical Equivalence: The Laws of Logic z Logically equivalent examples z p ⇔ (p∨p) z z ¬¬p ⇔ p z z z 2009 Spring “1+1=2” ⇔ “1+1=2 or 1+1=2” “He did not not do it” ⇔ “He did it” ¬(p∧q) (p∧q) ⇔ ¬p∨¬q p∨ q “If p then not p” ⇔ “not p” Discrete Mathematics – CH2 18 The Laws of Logic (1/2) 1) Law off Double bl Negation i : ¬¬p ⇔ p ⎧¬( p ∨ q ) ⇔ ¬p ∧ ¬q 2)DeMorgan DeMorgan' s Laws : ⎨ ⎩¬( p ∧ q ) ⇔ ¬p ∨ ¬q ⎧p ∨ q ⇔ q ∨ p 3) Commutative Laws : ⎨ ⎩p ∧ q ⇔ q ∧ p ⎧ p ∨ (q ∨ r ) ⇔ ( p ∨ q) ∨ r 4) Associative Laws : ⎨ ⎩ p ∧ (q ∧ r ) ⇔ ( p ∧ q) ∧ r ⎧ p ∨ (q ∧ r ) ⇔ ( p ∨ q) ∧ ( p ∨ r ) 5) Distributive Laws : ⎨ ⎩ p ∧ (q ∨ r ) ⇔ ( p ∧ q) ∨ ( p ∧ r ) 2009 Spring Discrete Mathematics – CH2 19 The Laws of Logic (2/2) ⎧p ∨ p ⇔ p 6) Idempotent Laws : ⎨ ⎩p ∧ p ⇔ p ⎧ p ∨ F0 ⇔ p 7) Identity Laws : ⎨ ⎩ p ∧ T0 ⇔ p ⎧ p ∨ ¬p ⇔ T0 8) Inverse Laws : ⎨ ⎩ p ∧ ¬p ⇔ F0 ⎧ p ∨ T0 ⇔ T0 9) Domination Laws : ⎨ ⎩ p ∧ F0 ⇔ F0 ⎧ p ∨ ( p ∧ q) ⇔ p 10) Absorption Laws : ⎨ ⎩ p ∧ ( p ∨ q) ⇔ p 2009 Spring Discrete Mathematics – CH2 20 2.2 Logical Equivalence: The Laws of Logic z ¬( p ∧ q ) ⇔ ¬p ∨ ¬q DeMorgan’s Laws: ¬( p ∨ q ) ⇔ ¬p ∧ ¬q p q 0 0 1 1 0 1 0 1 2009 Spring pרq ¬(p רq) 0 0 0 1 1 1 1 0 ¬p ¬q ¬ p ∨ ¬ q p ∨ q ¬(p ∨ q) 1 1 0 0 1 0 1 0 1 1 1 0 0 1 1 1 Discrete Mathematics – CH2 1 0 0 0 ¬p ¬רq 1 0 0 0 21 The Principle of Duality z Definition 2.3: Let s be a statement. Dual of s, denoted sd, is the statement obtained from s by replacing each occurrence of ∧ and ∨ by b ∨ andd ∧, respectively, ti l andd eachh occurrence off T0 andd F0 by b F0 and T0, respectively. z E.g. g s: ( p ∧ ¬q ) ∨ ( r ∧ T0 ) s d : ( p ∨ ¬q ) ∧ ( r ∨ F0 ) z Keep negation! The Principle of Duality: Let s and t be statements that contain no logical connectives other than ∧ and ∨. If s ⇔ t , then th sd ⇔ td. 2009 Spring Discrete Mathematics – CH2 22 Substitution Rules z Rule 1: Suppose that the compound statement P is a tautology. z z If p is a primitive statement that appears in P and we replace each occurrence off p by b the th same statement t t t q, then th th the resulting lti compound statement P1.is also a tautology. Rule 2: Let P be a compound p statement where p is an arbitraryy statement that appears in P, and let q be a statement such that q ⇔ p. z 2009 Spring Suppose that S h iin P we replace l one or more occurrences off p by b q. Then this replacement yields the compound statement P1. Under these circumstances P1 ⇔ P. Discrete Mathematics – CH2 23 Simplification of Compound Statements z Ex 2.10: z From the first of DeMorgan’s Laws P: ¬ (p ∨ q) ↔(¬p ∧¬q) is a tautology z from the first substitution rule P1: ¬ [(r ∧ s) ∨ q] ↔ [¬(r ∧ s) ∧ ¬q] is also a tautology z replace each occurrence of q by t →u P2: ¬ [(r ∧ s) ∨ (t →u)] ↔[¬ (r ∧ s) ∧ ¬ (t →u)] 2009 Spring Discrete Mathematics – CH2 24 Simplification of Compound Statements z Ex 2.11: z P: (p → q) →r, and because (p → q) ⇔ ¬p ∨ q z from the second substitution rule if P1: (¬p ∨ q ) →r, then P1 ⇔ P z Check 2.11 (b) for another example using the second rule z 2009 Spring p →(p ∨ q), ¬ ¬ p ⇔ p Discrete Mathematics – CH2 25 Simplification of Compound Statements Ex 2.12: z z 1. 2. 3 3. 4. 5. 2009 Spring Negate and simplify the compound statement (p ∨ q) → r (p ∨ q) → r ⇔ ¬(p ∨ q) ∨ r (First substitution rule) ¬[(p ∨ q) → r] ⇔ ¬[¬(p ∨ q) ∨ r] (Negating) ¬[¬(p ∨ q) ∨ r] ⇔ ¬¬(p ∨ q) ∧ ¬r (DeMorgan (DeMorgan’ss Laws) ¬¬(p ∨ q) ∧ ¬r ⇔ (p ∨ q) ∧ ¬r (Law of double Negation) ¬[(p [(p ∨ q) → r]] ⇔ (p ∨ q) ∧ ¬r Discrete Mathematics – CH2 26 Simplification of Compound Statements z Ex 2.13: z p: Joan goes to Lake George q: Mary pays for Joan’s shopping spree. p → q: If Joan goes to Lake George, then Mary will pay for Joan’ss shopping spree. Joan The negation of p → q : z z z z z 2009 Spring One way: ¬ (p → q). It is not the case that if Joan goes to Lake G George, th then M Mary will ill pay for f Joan’s J ’ shopping h i spree. Another way: p ∧ ¬ q. Joan goes to Lake George, but Mary does not pay for Joan’s shopping spree. ¬( p → q ) ⇔ ¬(¬p ∨ q) ⇔ ¬¬p ∧ ¬q ⇔ p ∧ ¬q Discrete Mathematics – CH2 27 Relevant Statements to Implication p Statement z p q p→q ¬q→¬p q→p ¬p→¬q 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 Ex 2.15: p → q ⇔ (¬ q → ¬ p) z q → p ⇔ (¬ ( p → ¬ q)) Contrapositive of p → q : ¬ q → ¬ p Converse of p → q : q → p Inverse of p → q : ¬ p → ¬ q z z z z 2009 Spring Discrete Mathematics – CH2 28 Simplification of Compound Statements z Ex 2.16: How to simply express the compound statement (p ∨ q) ∧ ¬(¬p ∧ q)? ( p ∨ q ) ∧ ¬(¬p ∧ q ) Reasons ⇔ ( p ∨ q ) ∧ (¬¬p ∨ ¬q ) (DeMorgan' s Law) ⇔ ( p ∨ q ) ∧ ( p ∨ ¬q ) ((Law of Double Negation) g ) 2009 Spring ⇔ p ∨ ( q ∧ ¬q ) (Distributive Law of ∨ over ∧) ⇔ p ∨ F0 (Inverse Law) ⇔p (Identity Law) Discrete Mathematics – CH2 29 Simplification of Compound Statements z Ex 22.18: 18: A switching network is made up of wires and switches connecting two terminals T1 and T2 • p p p q t ¬tt T1 r ¬q p • • T2 T1 r • t T2 r ¬q ( p ∨ q ∨ r ) ∧ ( p ∨ t ∨ ¬q ) ∧ ( p ∨ ¬t ∨ r ) ⇔ p ∨ [r ∧ (t ∨ ¬q )] Practice! P i ! 2009 Spring Discrete Mathematics – CH2 30 Simplification of Compound Statements (p ∨ q ∨ r) ∧ (p ∨ t ∨ ¬q) ∧ (p ∨ ¬t ∨ r) ⇔ p ∨ [(q ∨ r) ∧ (t ∨ ¬q) ∧ (¬t ∨ r)] ⇔ p ∨ [((q ∧ ¬t ) ∨ r) ∧ (t ∨ ¬q)] ⇔ p ∨ [((q ∧ ¬t ) ∨ r) ∧ ¬(¬t ∧ q)] ⇔ p ∨ [((q ∧ ¬tt ) ∧ ¬(¬t ( t ∧ q)) ∨ ((¬(¬t ( t ∧ q) ∧ r)] ⇔ p ∨ [F0 ∨ (¬(¬t ∧ q) ∧ r)] p ⇔ p ∨ [r ∧ (t ∨ ¬q)] • T1 t • T2 r ¬q 2009 Spring Discrete Mathematics – CH2 31 2.3 Logic Implication: Rules of Inference z z Argument: ( p1 ∧ p2 Premiss: p1, p2,…, pn Conclusion: q ∧⋅⋅⋅∧ pn ) → q Ex 2.19: z z z z z z z 2009 Spring p: Roger studies studies. q: Roger plays racketball. r: Roger passes discrete mathematics. p1 : p → r p1: Roger studies, then he will pass discrete mathematics. p 2 : ¬q → p p2: If Roger don don’tt play racketball racketball, then he he’ll ll study. study p3 : ¬ r p3: Roger failed discrete mathematics. Determine whether the argument ( p1 ∧ p 2 ∧ p3 ) → q is valid. Discrete Mathematics – CH2 32 Logic Implication: Rules of Inference z z Examine the truth table for the implication p [( p → r ) ∧ (¬q → p ) ∧ ¬r ] → q p1 p2 p3 (p1 רp p2רp p3) → q p q r p→r ¬q → p ¬r [(p → r )¬( רq → p) ¬רr] → q 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 1 1 1 1 0 1 0 1 0 0 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 ( p1 ∧ p2 ∧ p3 ) → q 2009 Spring p1 : p → r p 2 : ¬q → p p3 : ¬ r is a valid argument. argument Discrete Mathematics – CH2 33 Logic Implication: Rules of Inference z Definition 2.4: p logically implies q, p ⇒ q. z z If p, q are arbitrary statements such that p → q is a tautology. Rules of inference z z Use instead of constructing the huge truth table Using these techniques will enable us to consider only the cases wherein all the premises are true. Development of a step-by-step validation of how the concl sion q logically conclusion logicall follows follo s from the premises p1, p2,…, pn in an implication of the form ( p1 ∧ p2 2009 Spring ∧⋅⋅⋅∧ pn ) → q Discrete Mathematics – CH2 34 Logic Implication: Rules of Inference z Rule of Detachment (分離) z z z Modus Ponens: the method of affirming Example: [ p ∧ ( p → q)] → q p q p→q p ( רp → q) [p ( רp → q)] → q 0 0 1 1 0 1 0 1 1 1 0 1 0 0 0 1 1 1 1 1 Tabular form: therefore 2009 Spring p p→q ∴q Discrete Mathematics – CH2 35 Logic Implication: Rules of Inference z Law of the Syllogism (演繹推理) z Tabular form: p→q q→r ∴p→r z Modus Tollens: Method of Denying z Tabular form: p→q ¬q ∴ ¬p 2009 Spring Discrete Mathematics – CH2 36 Rules of Inference 2009 Spring Discrete Mathematics – CH2 37 Logic Implication: Rules of Inference Ex 2.30: 2009 Spring Discrete Mathematics – CH2 38 Logic Implication: Rules of Inference z Ex 2.31: z z z z If the band could not play rock music or the refreshments were not delivered on time, then the New Year’s party would have been cancelled and Alicia would have been angry. If the party were cancelled, then refunds would have had to be made. No refunds were made. Therefore the band could play rock music. p: The band could play rock music. q: The refreshments were delivered on time r: The New Year’s party was cancelled . s: Alicia was angry. t: Refunds had to be made made. 2009 Spring Discrete Mathematics – CH2 (¬ p ∨ ¬q ) → ( r ∧ s ) r→t ¬t ∴p 39 Logic Implication: Rules of Inference ( ¬p ∨ ¬q ) → ( r ∧ s ) r →t Steps (1) r → t Reasons Premise (2) ¬t (3) ¬r (4) ¬r ∨ ¬s Premise (1) and (2), and Method of Denying (3) andd Rule R l off Disjunctiv Di j i e Amplificat A lifi ion i (5) ¬(r ∧ s ) (6) (¬p ∨ ¬q) → (r ∧ s ) (4) and DeMorgan' s Laws Premise (7) ¬(¬p ∨ ¬q ) (8) p ∧ q (6) and (5), and Method of Denying (7) , DeMorgan' s Laws, and Law of Double Negation (9) ∴ p (8) and Rule of Conjunctive Simplification 2009 Spring Discrete Mathematics – CH2 ¬t ∴p 40 Proof by Contradiction z Ex 2.35: z z 2009 Spring Is the right argument valid or invalid? Let conclusion ¬p be false (the argument invalid) ⇒ p is true ⇒ q is true ⇒ s is true ⇒ r is false ⇒ r is i true t (¬p∨r, ( now p is i true) t ) ⇒ contradiction ⇒ argument is valid Discrete Mathematics – CH2 41 Interesting examples z A. 考前猜題 z z z z 習題A 1, A 習題A-1 A-2兩題中必出一題 2兩題中必出 題, 但不會兩題都 出 習題A-2和A-3兩題要嘛都出 習題A 2和A 3兩題要嘛都出, 要嘛都不出 如果不出A-1, 也不會出A-3 請問哪 題會出哪 題不會出? 請問哪一題會出哪一題不會出? A-1出, A-2不出, A-3不出 2009 Spring Discrete Mathematics – CH2 42 Interesting examples z B. 誰是兇手 z 角頭老大A死了, 警方請來他幫派裡兩位二哥 角頭老大A死了 級人物B和C來協助瞭解死因。 z z z 警方不知B C供詞真偽 但可以確定的是 警方不知B,C供詞真偽, z z z z 2009 Spring B說: 如果A被謀殺, 那肯定是C幹的 C說: 如果A不是自殺, 那就是被謀殺 A死因只有三種: 意外, 自殺和謀殺 如果B和C都沒有說謊 那A就死於 如果B和C都沒有說謊, 那A就死於一場意外 場意外 如果B和C兩人中有一人說謊, 那麼A就不是死於 意外 請問A的死因為何? Ans: 謀殺, B說謊 Discrete Mathematics – CH2 43 2.4 The Use of Quantifiers z Definition 2.5: A declarative sentence is an open statement if z z z z z z it contains one or more variables, and it is not a statement, but it becomes a statement when the variables in it are replaced by certain allowable choices. E.g., z p(x): The number x+2 is an even integer integer. z q(x, y): The number y+2, x - y, and x+2y are even integers. z For some x, p(x). For some x, y, q(x, y). z For some x, x ¬p(x). p(x) For some x, x y, y ¬q(x, q(x y). y) Existential quantifier: ∃x , “for some x”, “for at least one x” or “th exists “there i t an x suchh that” th t” Universal quantifier: ∀x , “for all x”, “for any x” “for every x” or “for each x such that” 2009 Spring Discrete Mathematics – CH2 44 The Use of Quantifiers z Ex 2.36: z z z z 2009 Spring 2 p ( x ) : x ≥ 0 , q ( x ) : x ≥0 Given the open statements r ( x) : x 2 − 3 x − 4 = 0, s ( x) : x 2 − 3 > 0 The statement ∀x [ p ( x ) → q ( x )] is true z For every real number x, if x ≥ 0, then x2 ≥ 0. z Every nonnegative real number has a nonnegative square. z The square of any nonnegative real number is a nonnegative reall number. b z All nonnegative real numbers have nonnegative squares. The statement ∃x [ p ( x) ∧ r ( x)] is true. The statement ∀x [ q ( x ) → s ( x )] is false. Discrete Mathematics – CH2 Counterexample! 45 Summarization ¬∃x: P(x) (x) v.s. ∃x: ¬P(x) (x) ¬∀x: P(x) v.s. ∀x: ¬P(x) 2009 Spring Discrete Mathematics – CH2 46 The Use of Quantifiers z For open statements p(x), q(x), the universally quantified statement ∀x[p(x) → q(x)], we define z z z 2009 Spring Contrapositive: ∀x[¬q(x) → ¬p(x)] Converse: ∀x[q(x) → p(x)] Inverse: ∀x[¬p(x) → ¬q(x)] Discrete Mathematics – CH2 47 Logical g Equivalence q and Implication p for Quantified Statements ∃x [ p ( x) ∧ q ( x)] ⇒ [∃x p ( x) ∧ ∃x q ( x)] ∃x [ p ( x) ∨ q ( x)] ⇔ [∃x p ( x) ∨ ∃x q ( x)] ∀x [ p ( x) ∧ q ( x)] ⇔ [∀x p ( x) ∧ ∀x q ( x)] [∀x p ( x) ∨ ∀x q ( x)] ⇒ ∀x [ p ( x) ∨ q ( x)] 2009 Spring Discrete Mathematics – CH2 48 The Use of Quantifiers z Rules for negating statements with one quantifier ¬[∀x p ( x )] ⇔ ∃x¬p ( x ) ¬[∃x p ( x )] ⇔ ∀x¬p ( x ) ¬[∀x ¬p ( x )] ⇔ ∃x¬¬ p ( x ) ⇔ ∃x p ( x ) ¬[∃x ¬p ( x )] ⇔ ∀x¬¬ p ( x ) ⇔ ∀x p ( x ) 2009 Spring Discrete Mathematics – CH2 49 Commuting Quantifiers Identical quantifiers commute: ∃x∃y: P(x,y) ⇔ ∃y∃x: P(x,y) and ∀x∀y: P(x,y) P(x y) ⇔ ∀y∀x: P(x,y) P(x y) But non-identical ones do not, see: ∃x∀y: y x=y y v.s. ∀y∃x: y x=yy 2009 Spring Discrete Mathematics – CH2 50 The Use of Quantifiers z Ex 2.49: z What is the negation of the following statement? ∀x ∃y [( p ( x, y ) ∧ q ( x, y )) → r ( x, y )] ¬[∀x ∃y [( p ( x, y ) ∧ q ( x, y )) → r ( x, y )]] ⇔ ∃x [¬∃ y [( p ( x, y ) ∧ q ( x, y )) → r ( x, y )]] ⇔ ∃x ∀y¬ [( p ( x, y ) ∧ q ( x, y )) → r ( x, y )] ⇔ ∃x ∀y¬ [¬[ p ( x, y ) ∧ q ( x, y )] ∨ r ( x, y )] ⇔ ∃x ∀y[[¬ ¬[ p ( x, y ) ∧ q ( x, y )] ∧ ¬r ( x, y )] ⇔ ∃x ∀y[ p ( x, y ) ∧ q ( x, y )] ∧ ¬r ( x, y )] 2009 Spring Discrete Mathematics – CH2 51 2.5 Quantifiers,, Definitions,, and the Proofs of Theorems z Ex 2.52: z z show that for all n, n=2, 4, 6,…,24, 26, we can write n as the sum off at most three h perfect f squares. Exhaustion method 2=1+1 4 4 4=4 6=4+1+1 8=4+4 8 4+4 2009 Spring 10=9+1 12 4 4 4 12=4+4++4 14=9+4+1 16=16 16 16 18=16+1+1 20=16+4 22 9 9 4 22=9+9+4 24=16+4+4 26=25+1 26 25+1 Discrete Mathematics – CH2 52 Quantifiers,, Definitions,, and the Proofs of Theorems z The R Th Rule l off U Universal i lS Specification: ifi ti If an open statement becomes true for all replacements by the members in a given universe, then that open statement is true for each specific individual member in that universe. if ∀x p (x) is true, then p ( x ) is true for each a in the universe. z 2009 Spring Eg E.g. ∀x [ m( x ) → c ( x )] m(a ) ∴ c (a (a) Steps Reasons (1) ∀x[m( x) → c( x)] Premise (2) m(a) (3) m(a) → c(a) Premise (1) and the rule of Universal Specification (4) ∴ c(a) ( ) ((2)) and ((3)) and the Rule of Detachment Discrete Mathematics – CH2 53 Quantifiers,, Definitions,, and the Proofs of Theorems z The R Th Rule l off U Universal i l Generalization: G li ti If an open statement p(x) ( ) is proved to be true when x is replaced by any arbitrarily chosen element c from our universe, then the universally quantified statement ∀x p(x) is true. Furthermore, the rule extends beyond a single variable. z Ex 2.56: ∀x [ p ( x) ∨ q ( x)] ∀x [(¬p ( x) ∧ q ( x)) → r ( x)] ∴ ∀x [¬r ( x) → p ( x)] 2009 Spring [¬r ( x) → ¬(¬p ( x) ∧ q ( x))] [¬r ( x) → p ( x) ∨ ¬q ( x))] Discrete Mathematics – CH2 54 Quantifiers, Definitions, and the Proofs of Theorems ∀x [ p( x) ∨ q( x)] z 2009 Spring Ex 2.56: ∀x [(¬p ( x) ∧ q ( x)) → r ( x)] ∴ ∀x [¬r ( x) → p ( x)] Discrete Mathematics – CH2 55 Quantifiers,, Definitions,, and the Proofs of Theorems z Argument ∀x [( j ( x ) ∨ s ( x )) → ¬p ( x )] p (m) ∴ ¬s (m (m) z Establish the validity of this argument • No junior or senior is enrolled in a physical education class. • Mary is enrolled in a physical education class. • Thus Mary is not a senior. Steps Reasons 1) ∀x [ j ( x) ∨ s ( x) → ¬p( x)] Premise 2) p (m) Premise 3) j (m) ∨ s (m) → ¬p (m) (1) and Rule of Universal Specification 4) p (m) → ¬( j (m) ∨ s (m)) (3), (q → t ) ⇔ (¬t → ¬q) and Law of Double Negation 5) p (m) → (¬j (m) ∧ ¬s (m)) (4) and DeMorgan' s Law 6) ¬ j ( m ) ∧ ¬ s ( m ) (2) and (5) and Rule of Detachment 7 ) ∴ ¬s ( m ) (6) and Rule of Conjunctive Simplification 2009 Spring Discrete Mathematics – CH2 56 Quantifiers,, Definitions,, and the Proofs of Theorems z D fi iti 2.8: Definition 28 z z Let n be an integer. We call n even if n is divisible by 2, that is, if there exists an integer r so that n = 2r. If n is not even, then we call n odd and find for this case that there exists an integer s where n = 2s +1. Theorem 2.2: 2 2: For all integers k and l, l if k and l are both odd, odd then k + l is even. z 1. 2. 3. 2009 Spring Proof Si Since k andd l are odd, dd we may write i k = 2a+1 2 1 andd l = 2b+1, 2b 1 for f some integers a, b. (the Rule of Universal Specification) Then k + l = (2a+1)+(2b+1) = 2(a+b+1), which hold for integers. (apply Commutative Commutative, Associative Associative, and Distributive Laws) Since a, b are integers, a+b+1 = c is an integer; with k + l = 2c, so k + l is even. Discrete Mathematics – CH2 57 Quantifiers,, Definitions,, and the Proofs of Theorems z Theorem 2.3: For all integers k and l, if k and l are both odd, then kl is also odd. z 2009 Spring Proof? Discrete Mathematics – CH2 58 Quantifiers,, Definitions,, and the Proofs of Theorems Theorem 2.4: If m is an even integer, then m + 7 is odd. z z 1. 2. 3. 2009 Spring Proof Since m is even, we have m = 2a for some integer a. Then m + 7 = 2a + 7 = 2a + 6 +1 = 2(a + 3) + 1. since a+3 is an integer, we know that m + 7 is odd. Suppose pp that m + 7 is not odd, hence even. Then m + 7 = 2b for some integer b, and m = 2b – 7 = 2b – 8 +1 = 2(b – 4) +1, where b – 4 is an integer. Hence m is odd. (contraposition method) Assume that m is even and m + 7 is also even. Then m + 7 even i li that implies th t m + 7 = 2c 2 for f some integer i t c. Consequently, C tl m = 2c 2 – 7 = 2c – 8 +1 = 2(c – 4) +1 with c – 4 an integer, so m is odd. Now we have contradiction. So the assumption is false (m + 7 is even), and we have m + 7 odd. (contradiction method) Assumption Result Derived Contraposition ¬q(m) ¬p(m) Contradiction p(m) and ¬q(m) F0 Discrete Mathematics – CH2 59 Proof Prove (p and ¬ q) is false Î prove (¬p or q) is true 2009 Spring Discrete Mathematics – CH2 60 Quantifiers,, Definitions,, and the Proofs of Theorems Theorem 2.5: z z z For all positive real numbers x and y, y if the product xy exceeds 25, then x > 5 or y > 5. Proof: (Contrapositive) z z 2009 Spring Suppose that 0 < x ≤ 5 and 0 < y ≤ 5 (¬q(x, y)) We find that 0 = 0.0 < x. y ≤ 5.5 = 25 (¬p(x, y)) Discrete Mathematics – CH2 61 Exercise z z z z z 2.1: 10 2 2: 16, 2.2: 16 20 2.3: 10 2.4: 14 2 5: 16 2.5: 2009 Spring Discrete Mathematics – CH2 62