Logic and Proof. Aiichiro Nakano

Transcription

1 Logic and Proof Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Department of Computer Science Department of Physics & Astronomy Department of Chemical Engineering & Materials Science Department of Biological Sciences University of Southern California rom K. H. Rosen, Discrete Mathematics and Its Applications (McGraw-Hill) Chapter 1

2 Logic Logic = mathematical foundation for: Objective reasoning ormal proofs Computation Proposition: A declarative sentence that is either true or false, but not both (Example) today is riday read this book imperative what time is it interrogative Predicate (propositional function): A statement involving variables (Example) P(x, y): Student x has studied subject y ruth value: rue () or false ()

3 Connectives (1) Logical operators to form composite propositions from existing propositions Negation p: not p ; inverts the truth value Conjunction p Ù q: p and q ; if both p and q are Disjunction p Ú q: p or q ; if both p and q are, otherwise Exclusive or p Å q: when exactly one of p and q is ruth table: Combinatorial enumeration of the truth values of composite propositions; 2 n rows for a n-proposition composite Unary p p Binary p q p Ù q p Ú q p Å q

4 Connectives (2) Implication: p q; when p is and q is, and otherwise hypothesis conclusion if p, then q ( q if p ) p implies q p is a sufficient condition for q p q p q Related implications Contrapositive: q p Converse: q p Inverse: p q Biconditional: p «q; if p and q has the same truth value q if and only if (iff) p p q p «q p is sufficient and necessary for q

5 Nested Connectives Use parenthesis (p Ù q) Ú r Precedence: No need to memorize, except for p Ù q is ( p) Ù q not (p Ù q) Operator Precedence 1 Ù 2 Ú 3 operate first «4 5 operate last

6 Logical Equivalences autology: A compound proposition that is always p º q: p and q are logically equivalent; p «q is a tautology Syntactically different composite propositions can have the same meaning (truth values) ruth table can be used to determine the equivalence (Example) equivalence of contrapositive p q º q p p q p q q p q p

7 Basic Logical Equivalences (1) Useful to simplify complex composite propositions p Ù º p p Ú º p p Ú º p Ù º p Ú p º p p Ù p º p ( p) º p Identity laws Domination laws Idempotent laws Double negation

8 Basic Logical Equivalences (2) Ù and Ú are commutative, associative, and distributive p Ù q º q Ù p p Ú q º q Ú p (p Ù q) Ù r º p Ù (q Ù r ) (p Ú q) Ú r º p Ú (q Ú r) p Ú (q Ù r) º (p Ú q ) Ù (p Ú r ) p Ù (q Ú r) º (p Ù q ) Ú (p Ù r ) Commutative laws Associative laws Distributive laws Associative laws make the following well defined p 1 Ù p 2 Ù... Ù p n if all are p 1 Ú p 2 Ú... Ú p n if one is

9 Basic Logical Equivalences (3) (p Ù q) º p Ú q (p Ú q) º p Ù q p Ú (p Ù q) º p p Ù (p Ú q) º p p Ú p º p Ù p º De Morgan s laws Absorption laws Negation laws Generalized De Morgan s theorem (p 1 Ù p 2 Ù... Ù p n ) º p 1 Ú p 2 Ú... Ú p n it is not the case that all are º one is (p 1 Ú p 2 Ú... Ú p n ) º p 1 Ù p 2 Ù... Ù p n it is not the case that one is º all are

10 Basic Logical Equivalences (4) Equivalences involving implication & biconditional in terms of other connectives p q º p Ú q p q º q p (contrapositive) p «q º (p q) Ù (q p)

11 Basic Logical Equivalences (5) Additional logical equivalences can be derived, combining other logical equivalences that have already been established (Example) Show that (p Ù q) (p Ú q) º i.e., (p Ù q) (p Ú q) is a tautology (p Ù q) (p Ú q) º (p Ù q) Ú (p Ú q) p q º p Ú q º ( p Ú q) Ú (p Ú q) De Morgan º p Ú q Ú p Ú q Associative law º ( p Ú p) Ú ( q Ú q) Associative law º Ú Negation laws º Domination law

12 Quantifiers Universe of discourse: he collection of values that a variable can take Universal quantifier, ": "xp(x), for all x P(x) P(x) is for all values of x in the universe of discourse (Example) Universe of discourse consists of all integers "x(x + 1 > x) is "x(x 2 > x) is (P(0) is, counterexample) Existential quantification, $: $xp(x), there is an x such that P(x) here exists an element x in the universe of discourse such that P(x) is (Example) Universe of discourse consists of all integers $x(x > 3) is (P(4) is for example)

13 Negating Quantifiers "xp(x) º $x P(x) it is not the case that P(x) is for all x º there exists x such that P(x) is (Example) Negation of all Americans eat cheeseburgers there is an American who do not eat cheeseburgers $xp(x) º "x P(x) it is not the case that there exists x such that P(x) is º P(x) is for all x (Example) Negation of there is an honest politician all politicians are dishonest hese equivalences are just De Morgan s theorems: (P(x 1 ) Ù P(x 2 ) Ù... Ù P(x n )) º P(x 1 ) Ú P(x 2 ) Ú... Ú P(x n ) (P(x 1 ) Ú P(x 2 ) Ú... Ú P(x n )) º P(x 1 ) Ù P(x 2 ) Ù... Ù P(x n )

14 Nesting Quantifiers 1. "x"yp(x,y): P(x,y) is for every pair x,y 2. $x$yp(x,y): there is a pair for which P(x,y) is 3. "x$yp(x,y): for every x there is a y for which P(x,y) is 4. $x"yp(x,y): there is a x, for which P(x,y) is for all y (Example) P(x, y) = x relies upon y "x($yp(x,y)): everyone has someone to rely on $x("yp(x,y)): there is a needy person who relies on everybody

15 Proof Axioms: Statements assumed to be, defining a mathematical structure heorem: A statement that can be shown to be Proof: Demonstration that a theorem is Rules of inference: Patterns to deduce a true statement (conclusion) from a set of other true statements (hypotheses) General form: If we know that a set of hypotheses are all, then a conclusion is hypothesis 1 (h 1 ) hypothesis 2 (h 2 )... \ (denotes therefore ) conclusion (c) Associated tautology: h 1 Ù h 2 Ù... c

16 Examples of Axioms Axioms of Euclidean geometry (~300BC) 1. Given two distinct points, one can draw one and only one line segment connecting these points. 2. Given two distinct points, one can draw one and only one circle centered at the first point and passing through the second one. 3. Any two right angles are equal. 4. Every line segment can be infinitely continued in either direction. 5. or any given line l and a point P not on that line one can draw one and only one line l 1 through P that will not intersect the original line l. Axioms of Riemann s geometry (1854) Euclid s axioms Given a straight line and a point not on the line, there are no straight lines through the point parallel to the original line. Axioms in Newtonian mechanics 1. An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force. 2. he acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object. 3. or every action, there is an equal and opposite reaction.

17 Rules of Inference Rule of Inference autology Name p. p (p Ú q) Addition \ p Ú q p Ù q. (p Ù q) p Simplification \ p p (p) Ù (q) p Ù q Conjunction q. \ p Ù q p [p Ù (p q)] q Modus ponens p q. \q q p q. \ p [ q Ù (p q)] p Modus tollens

18 Rules of Inference Rule of Inference autology Name p q [(p q) Ù (q r)] (p r) Hypothetical q r. syllogism \p r p Ú q p. \q p Ú q p Ú r. \q Ú r [(p Ú q) Ù p] q [(p Ú q) Ù ( p Ú r)] (q Ú r) Disjunctive syllogism Resolution

19 Rules of Inference for Quantified Statements Rule of Inference "xp(x). \P(c) for a particular c P(c) for an arbitrary c \"xp(x) $xp(x). \P(c) for some c P(c) for some c \$xp(x) Name Universal instantiation Universal generalization Existential instantiation Existential generalization

20 Methods of Proving heorems Proving implications p q: Direct proof: Assume p is, and use rules of inference to prove that q is Indirect proof: Prove its contrapositive; assume q, and prove p Proof by cases: Prove (p 1 Ú p 2 ) q by proving (p 1 q) and (p 1 q) Based on [(p 1 Ú p 2 ) q] º [(p 1 q) Ù (p 2 q)]

21 Sample Direct Proof Write your answer to a homework at this level of readability!

22 Proof by Contradiction Proving p: Prove p (q Ù q) Since (q Ù q) º (contradiction, by negation law), p his is only, if p = (i.e., p = ) (Example) Prove that p: Ö2 is irrational Assume p: Ö2 is rational (i.e., Ö2 = a/b, where integers a & b have no common factors). It follows that 2 = a 2 /b 2, hence 2b 2 = a 2. his means a 2 is even, which implies a is even. urthermore, since a is even, a = 2c for some integer c. hus 2b 2 = 4c 2, so b 2 = 2c 2. his means b 2 is even, which implies b is even. It has been shown that p Ö2 = a/b, where integers a & b have no common factors Ù 2 divides both a & b.

23 Existence Proofs Existence proof: Proving $xp(x) Constructive proof: ind an element a such that P(a) is Nonconstructive proof: or example, proof by contradiction prove that its negation leads to contradiction