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## Discrete Mathematics Lecture 2.

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**Discrete Mathematics Lecture 2.**Dr.Bassant Mohamed El-Bagoury dr.bassantai@gmail.com Module Logic (part 2 --- proof methods)**Outline**• 1. Mathematical Reasoning • 2. Arguments Examples – Predicate Logic • 3. Rules of Inference – Knowledge Engineering • 4. Rules of Inference for Quantifiers • 4. Methods for Theorem Proving**Mathematical Reasoning**• We need mathematical reasoning to • determine whether a mathematical argument is correct or incorrect and • construct mathematical arguments. • Mathematical reasoning is not only important for conducting proofs and program verification, but also for artificial intelligence systems (drawing inferences).**Arguments**• Example: • Gary is either intelligent or a good actor. • If Gary is intelligent, then he can count from 1 to 10. • Gary can only count from 1 to 2. • Therefore, Gary is a good actor. • i: “Gary is intelligent.” • a: “Gary is a good actor.” • c: “Gary can count from 1 to 10.”**Arguments**• i: “Gary is intelligent.”a: “Gary is a good actor.”c: “Gary can count from 1 to 10.” • Step 1: c Hypothesis • Step 2: i c Hypothesis • Step 3: i Modus Tollens Steps 1 & 2 • Step 4: a i Hypothesis • Step 5: a Disjunctive SyllogismSteps 3 & 4 • Conclusion: a (“Gary is a good actor.”)**Arguments**• Another example: • “If it rains today, then we will not have a barbeque today. If we do not have a barbeque today, then we will have a barbeque tomorrow.Therefore, if it rains today, then we will have a barbeque tomorrow.” • This is a valid argument: If its hypotheses are true, then its conclusion is also true.**AI research × Software Engineering**Intelligent Behaviour Requirements Specification Functional Specification Knowledge Level System Implementation System Level Symbol Level**A small Expert System ( Inference System ) of rules for**analysis of automotive problems. Rule 1:if the engine is getting gas, and the engine will turn over, then the problem is spark plugs. Rule 2:if the engine does not turn over, and the lights do not come on then the problem is battery or cables. Rule 3:if the engine does not turn over, and the lights do come on then the problem is the starter motor. Rule 4:if there is gas in the fuel tank, and there is gas in the carburetor then the engine is getting gas.**Fig The Inference System at the start of a consultation in**the car diagnostic example.**The and/or graph searched in the car diagnosis example, with**the conclusion of Inference Rule 4 matching the first premise of Inference Rule 1.**The following dialogue begins with the computer asking the**user about the goals present in working memory. Gas in fuel tank? Yes Gas in carburetor? Yes Engine will turn over? Why It has been established that: 1. The engine is getting gas, 2. The engine will turn over, Then the problem is the spark plugs. How the engine is getting gas This follows from rule 4: if gas in fuel tank, and gas in carburetor then engine is getting gas. gas in fuel tank was given by the user gas in carburetor was given by the user**Terminology**• An axiom is a basic assumption about mathematical structures that needs no proof. • We can use a proof to demonstrate that a particular statement is true. A proof consists of a sequence of statements that form an argument. • The steps that connect the statements in such a sequence are the rules of inference. • Cases of incorrect reasoning are called fallacies. • A theorem is a statement that can be shown to be true.**Terminology**• A lemma is a simple theorem used as an intermediate result in the proof of another theorem. • A corollary is a proposition that follows directly from a theorem that has been proved. • A conjecture is a statement whose truth value is unknown. Once it is proven, it becomes a theorem.**Arguments**• Just like a rule of inference, an argument consists of one or more hypotheses and a conclusion. • We say that an argument is valid, if whenever all its hypotheses are true, its conclusion is also true. • However, if any hypothesis is false, even a valid argument can lead to an incorrect conclusion.**Arguments**• Example: • “If 101 is divisible by 3, then 1012 is divisible by 9. 101 is divisible by 3. Consequently, 1012 is divisible by 9.” • Although the argument is valid, its conclusion is incorrect, because one of the hypotheses is false (“101 is divisible by 3.”). • If in the above argument we replace 101 with 102, we could correctly conclude that 1022 is divisible by 9.**Theorems, proofs, and rules of inference**• When is a mathematical argument (or “proof”) correct? • What techniques can we use to construct a mathematical argument? • Theorem– statement that can be shown to be true. • Axioms or postulates or premises– statements which are given and assumed to be true. • Proof– sequence of statements, a valid Argument, to show that a theorem is true. • Rules of Inference– rules used in a proof to draw conclusions from assertions known to be true.**Valid Arguments**(reminder) • Recall: • An argument is a sequence of propositions. The final proposition is called the conclusion of the argument while the other propositions are called the premises or hypotheses of the argument. • An Argument is valid whenever the truth of all its premises implies the truth of its conclusion. • How to show that q logically follows from the hypotheses (p1 p2 …pn)? Show that (p1 p2 …pn) q is a tautology One can use the rules of inference to show the validity of an argument. Vacuous proof - if one of the premises is false then (p1 p2 …pn) q is vacuously True, since False implies anything.**Methods of Proof**• 1) Direct Proof • 2) Proof by Contraposition • 3) Proof by Contradiction • 4) Proof of Equivalences • 5) Proof by Cases • 6) Existence Proofs • 7) Counterexamples**1) Direct Proof**• Proof statement : p q • by: • Assume p • From p derive q.**((M C) (D C) (D S) (M)) **S ? Direct proof --- Example 1 • Here’s what you know: Mary is a Math major or a CS major. If Mary does not like discrete math, she is not a CS major. If Mary likes discrete math, she is smart. Mary is not a math major. • Can you conclude Mary is smart? Let M - Mary is a Math major C – Mary is a CS major D – Mary likes discrete math S – Mary is smart Informally, what’s the inference chain of reasoning? M C D C D S M**((M C) (D C) (D S) (M)) **S ? • In general, to prove p q, assume p and show that q follows.**See Table 1, p. 66, Rosen.**Reminder: Propositional logic Rules of Inference or Method of Proof Subsumes MP**5. C**6. D 7. S Mary is smart! Example 1 - direct proof • 1. M C Given (premise) • 2. D C Given • 3. D S Given • 4. M Given DS (disjunctive syllogism; 1,4) MT (modus tollens; 2,5) MP (modus ponens; 3,6) QED QED or Q.E.D. --- quod erat demonstrandum**Direct Proof --- Example 2**• Theorem: • If n is odd integer, then n2 is odd. • Looks plausible, but… • How do we proceed? How do we prove this? • Start with • Definition: An integer is even if there exists an integer k such that n = 2k, • and n is odd if there exists an integer k such that n = 2k+1. • Properties: An integer is even or odd; and no integer is • both even and odd. (aside: would require proof.)**Example 2: Direct Proof**• Theorem: • (n) P(n) Q(n), • where P(n) is “n is an odd integer” and Q(n) is “n2 is odd.” • We will show P(n) Q(n)**Theorem:**• If n is odd integer, then n2 is odd. • Proof: • Let P --- “n is odd integer” • Q --- “n2 is odd” • we want to show that P Q • Assume P, i.e., n is odd. • By definition n = 2k + 1, where k is some integer. • Therefore n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2 (2k2 + 2k ) + 1, • which is by definition is an odd number (use k’ = (2k2 + 2k ) ). • QED Proof strategy hint: Go back to definitions of concepts and start by trying direct proof.**The Foundations: Logic and Proofs**Chapter 1**Propositional Logic**Proposition is a declarative statement that is either true of false • Baton Rouge is the capital of Louisiana True • Toronto is the capital of Canada False • 1+1=2 True • 2+2=3 False Statements which are not propositions: • What time is it? • x+1 = 2**Negation:**truth table**Conjunction:**truth table**Disjunction:**truth table**Exclusive-or:**one or the other but not both truth table**(hypothesis)**(conclusion) Conditional statement: if p then q p implies q q follows from p p only if q p is sufficient for q truth table**Conditional statement:**equivalent (same truth table) Contrapositive: Converse: equivalent Inverse:**Biconditional statement:**p if and only if q p iff q If p then q and conversely p is necessary and sufficient for q truth table**Compound propositions**Precedence of operators higher lower**Propositional Equivalences**Compound proposition Tautology: always true Contradiction: always false tautology contradiction Contingency: not a tautology and not a contradiction**Rules of Inference**If you have a current password, then you can log onto the network You have a current password Therefore, you can log onto the network Modus Ponens Valid argument: if premises are true then conclusion is true**Modus Ponens**If and then**Rules of Inference**Modus Ponens Modus Tollens Hypothetical Syllogism Disjunctive Syllogism**Rules of Inference**Addition Simplification Conjunction Resolution**It is below freezing now**Therefore, it is either below freezing or raining now Addition**It is below freezing**and raining now Therefore, it is below freezing now Simplification**If it rains today**then we will not have a barbecue today If we do not have a barbecue today then we will have a barbecue tomorrow Therefore, if it rains today then we will have a barbecue tomorrow Hypothetical Syllogism**it is not snowing**or Jasmine is skiing It is snowing or Bart is playing hockey Therefore, Jasmine is skiing or Bart is playing hockey Resolution