4 Interlude: A review of logic and proofs

Transcription

1 4 Interlude: A review of logic and proofs 4.1 Logic Propositions and logical connectives A proposition is a declarative sentence that is either true (T or false (F, but not both. Example 4.1 Examples of propositions: It rains every Monday. Whenever I am happy, I sing. If x 1, then x 2 x. x > y if and only if x 2 > y 2. Compound propositions can be formed from one or more propositions using logical operators. The following are the most important compound propositions. Negation: p, not p. True if and only if p is false. Conjunction: p q, p and q. True if and only if both p and q are true. Disjunction: p q, p or q (or both. True if and only if at least one of p and q is true. Implication: p q, if p, then q. (Also read: p implies q, p is sufficient for q, p only if q, q is a necessary condition for p. False if and only if p is true and q is false. Biconditional: p q, p if and only if q. (Also read: p is necessary and sufficient for q. True if and only if p and q are both true or both false. A truth table displays the relationship between the truth value of a compound proposition and the truth values of the individual propositions within it. Carefully examine the truth tables of the compound propositions in the table below. Pay particular attention to the truth values of the implication! p q p p q p q p q p q T T F T T T T T F F F T F F F T T F T T F F F T F F T T 39

2 4.1.2 Logical equivalence A compound proposition is called a tautology if it is true no matter what the truth values of the propositions within it; contradiction if it is false no matter what the truth values of the propositions within it; contingency if it can assume both true and false values. Exercise 4.2 Use truth tables to show that p p is a tautology and p p is a contradiction. Propositions p and q are called logically equivalent (denoted p q if p q is a tautology; that is, if p and q always assume the same truth value. Exercise 4.3 Some of the most important logical equivalences are shown in the table below. Here, T denotes a tautology and F denotes a contradiction. Use truth tables to prove these equivalences. Logical equivalence Name p T p Identity laws p F p p T T Domination laws p F F p p p Idempotent laws p p p ( p p Double negation law p q q p Commutative laws p q q p (p q r p (q r Associative laws (p q r p (q r p (q r (p q (p r Distributive laws p (q r (p q (p r (p q p q De Morgan s laws (p q p q p (p q p Absorption laws p (p q p p p T Negation laws p p F p q p q Implication and disjunction p q p q p q q p The contrapositive p q (p q (q p Biconditional and implication p q (p q ( p q Biconditional, conjunction, and disjunction 40

3 Exercise 4.4 Given an implication p q, the following important implications can be formed: q p, called the converse of p q; q p, called the contrapositive of p q; p q, called the inverse of p q. Using truth tables, prove that an implication is equivalent to its contrapositive, but not equivalent to its converse and inverse. Keep this result in mind whenever proving a theorem! Predicates and quantifiers A propositional function (or predicate P is a mapping that assigns to each value c in its domain (also called the universe of discourse a proposition P (c. Example 4.5 Let P (x denote the statement x > 3. As long as x is a variable (has no assigned value, this is not a proposition because it is neither true nor false. When x is assigned a value, say x = c, then P (c is a proposition, that is, a statement that is either true or false. For example, P (4 is the proposition 4 >3 (which is true, and P (2 is the proposition 2 > 3 (which is false. Another way to convert a propositional function into a proposition is to use quantifiers: The universal quantification of P (x is xp (x, read for all x (in the universe of discourse P (x (is true. The existential quantification of P (x is xp (x, read there exists x (in the universe of discourse such that P (x (is true. Carefully examine the truth values of the quantified propositions in the table below. Proposition When true? When false? xp (x P (c is true for all c in the domain P (c is false for some c in the domain xp (x P (c is true for some c in the domain P (c is false for all c in the domain xp (x P (c is false for some c in the domain P (c is true for all c in the domain x P (x xp (x P (c is false for all c in the domain P (c is true for some c in the domain x P (x 41

4 4.2 Methods of proof In this section, we look at what constitutes a correct mathematical argument that can be used in a proof of a theorem A theorem is a mathematical statement that can be shown to be true. A theorem is sometimes called a proposition, fact, or result. A lemma is a theorem whose main purpose is to help in the proof of another theorem. A corollary is a theorem that easily follows from another theorem. A conjecture is a proposition whose truth value is unknown; once proved (if true, it becomes a theorem. A proof is a mathematical argument that demonstrates that a theorem is true. An argument is a set of propositions in which one, called the conclusion, is claimed to follow from the others, called the hypotheses or premises (more about arguments later. The hypotheses of a proof may be axioms or postulates, which are the underlying assumptions about mathematical structures, the hypotheses of the theorem to be proved, or previously proved theorems. The steps of the proof are tied together by the rules of inference (to be discussed below, or correct reasoning Rules of inference Rules of inference are the means to draw conclusions from other assertions, thereby correctly tying together the steps of the proof. Each rule of inference for propositions (see the table below is simply a tautology. Rule of inference Tautology Name p p q p (p q Addition p q p (p q p Simplification p q ((p (q (p q Conjunction p q p p q q (p (p q q Modus ponens (law of detachment q p q p ( q (p q p Modus tollens p q q r p r ((p q (q r (p r Hypothetical syllogism p q p q ((p q ( p q Disjunctive syllogism p q p r q r ((p q ( p r (q r Resolution Exercise 4.6 Using truth tables, show that each rule of inference in the table above is a tautology. 42

5 4.2.2 Arguments As mentioned above, an argument is a set of propositions in which one (called the conclusion is claimed to follow from the others (called the hypotheses or premises. That is, an argument is a proposition of the form (p 1 p 2... p k c, where p 1, p 2,..., p k are the premises and c is the conclusion. This is also written as p 1 p 2. The argument (* is said to be valid if the proposition (p 1 p 2... p k c is a tautology; that is, if the conclusion c is true whenever the premises p 1, p 2,..., p k are all true. Validity of an argument can be verified by following the rules of inference or by using truth tables. Example 4.7 Show that the hypotheses It is not sunny this afternoon and it is colder than yesterday We will go swimming only if it is sunny If we do not go swimming, then we will take a canoe trip If we take a canoe trip, then we will be home by sunset lead to the conclusion We will be home by sunset. Exercise 4.8 Show that the hypotheses If you send me an message, then I will finish doing my homework If you do not send me an message, then I will go to bed early If I go to bed early, then I will wake up feeling refreshed lead to the conclusion If I do not finish doing my homework, then I will wake up feeling refreshed. The following rules of inference for quantified statements are used extensively in mathematical arguments. p k c ( Rule of inference xp (x P (c for an arbitrary 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 43

6 Example 4.9 What rules of inference are used in the following famous argument? All men are mortal. Socrates is a man. Therefore, Socrates is mortal. Exercise 4.10 What rules of inference are used in the following argument? No man is an island. Baffin is an island. Therefore, Baffin is not a man. Exercise 4.11 Show that the premises A student in this class has not read the notes Everybody in this class passed the midterm exam imply the conclusion Someone who passed the midterm exam has not read the notes. A fallacy is a common form of incorrect reasoning. Fallacies resemble rules of inference but are based on contingencies rather than tautologies. In order to write correct mathematical arguments, it is very important to recognize them. Here are the two most common fallacies: Fallacy of confirming the conclusion: ((p q q p Fallacy of denying the hypothesis: ((p q p q Example 4.12 Examine the truth tables of the propositions ((p q q p and ((p q p q to show that they are indeed contingencies. Example 4.13 Are the following arguments valid? If not, what fallacy has been committed? 1. If the weather is nice, we go skiing. We are skiing. Therefore, the weather is nice. 2. If the weather is nice, we go skiing. The weather is nice. Therefore, we are skiing. 3. If the weather is nice, we go skiing. We are not skiing. Therefore, the weather is not nice. 4. If the weather is nice, we go skiing. The weather is not nice. Therefore, we are not skiing Methods of proving theorems The table below lists some of the most important methods for proving theorems. Note that most theorems are written in the form of an implication: one needs to prove that p q is true (that is, that p implies q, and not that q itself is true. 44

7 Type of Proof To Prove Approach Direct proof p q Assume p is true. Using the rules of inference show q is true. Indirect proof p q Prove q p (the contrapositive. Vacuous proof p q Show p is false. Trivial proof p q Show q is true. Proof by p Show p F is true. contradiction Proof by cases (p 1 p 2... p k q Prove p i q for each i = 1, 2,..., k. Proof of equivalence p q Prove both p q and q p. Proof of equivalence p 1 p 2... p k Prove p 1 p 2, p 2 p 3,..., and p k p 1. Existence proof xp (x Find c such that P (c is true. constructive Existence proof xp (x Prove xp (x without finding c such that nonconstructive P (c is true (e.g. by contradiction. Uniqueness proof!xp (x Prove xp (x. Prove (P (x P (y (x = y. Proof by xp (x Find c such that P (c is false. counterexample Exercise 4.14 Give an example for each of the types of proofs in the table above Mistakes in proofs Here are some common errors made in mathematical proofs: Using incorrect rules of inference or committing a fallacy (in particular, the fallacy of denying the hypothesis or the fallacy of affirming the conclusion. Proving q p instead of p q. Using q p instead of p q as a hypothesis of the theorem. Circular reasoning (also called begging the question: To prove proposition p, a proposition equivalent to p is used. Committing an error in arithmetic or basic algebra (e.g. dividing by x where x might be 0. Omitting a case in a proof by cases. 45

8 4.2.5 Paul Elliott-Magwood s advice on writing proofs 1. If you assume what you are trying to prove then of course the result will follow (circular reasoning!. Unfortunately, this is not a valid proof. 2. An assumed statement which implies (through a sequence of logical statements a true statement is not necessarily true (that is, p q being true does not make p true this is the fallacy of affirming the hypothesis. However, an assumed statement which leads to a contradiction is false (that is, if p F is true, then p must be false. 3. In terms of style, try to avoid writing unproven statements, even if you later follow with the proof. If you do wish to do this, write them as a separate lemma or claim. 4. A proof must be a sequence of logical steps, each one logically following from the previous (using the rules of inference. Each step should use a single idea (i.e. precisely applying a definition, precisely applying a theorem, appealing to an earlier assumption, or making a single observation. A general discussion is not a proof. Also, I (Mateja would like to add that a proof is not an essay and you should not try to write down all that you know on the topic. Rather, present only the ingredients that (using correct logical reasoning lead from the assumptions to the conclusion. In other words, your proof should be as direct a route from p to q as possible, without de-tours. 5. Proofs must be rigorous and must take into account every possible case. 6. Definitions are important. Be sure to correctly use existing mathematical language and notation (i.e. two vertices being adjacent is not the same as connected. Do not make up new words or new notation unless you write a precise definition of your new word or notation. Paul also mentions that on Assignment 1, there were lots of difficulties with concepts 1, 2, and 3 in Question 2, and with concepts 4, 5, and 6 in Question Mathematical induction The table below gives templates for the most commonly used forms of mathematical induction. Here are some additional recommendations for writing induction proofs on graphs. (This is written for the basic form of induction; try to rewrite the induction step to fit the format of Strong Induction. Carefully choose the graph parameter on which to use the induction. This can be the number of vertices, number of edges, number of connected components,... Below, I ll call this the size of the graph. The statement to prove should be of the following form: P (n is true for all n n 0, 46

9 where P (n is a statement of the form P (n: property P holds for all graphs (in a given domain of size n. Basis of induction: Prove P (n 0 holds for all graphs of size n 0. Induction Step: Assume P (n holds for all graphs of size n, for some n n 0 (Induction Hypothesis. Take an arbitrary graph G of size n + 1. Now prove property P holds for G. To use the induction hypothesis, remove a vertex, edge, a connected component,... to obtain a graph G of size n from the graph G. Be sure that G lies in the same domain as G so that the induction hypothesis really applies to it. Type of Statement You Must Prove Induction to Prove Mathematical P (n for all Basis Step: P (1 Induction n Z + Induction Step: P (k P (k + 1 for all k Z + Mathematical P (n for all Basis Step: P (n 0 Induction n Z, n n 0 Induction Step: P (k P (k + 1 for all k Z, k n 0 a generalization Strong P (n for all Basis Step: P (1 Induction n Z + Induction Step: (P (1 P (2... P (k P (k + 1 for all k Z + Strong P (n for all Basis Step: P (n 0 Induction n Z, n n 0 Induction Step: (P (n 0 P (n P (k P (k + 1 a generalization for all k Z, k n 0 References for Chapter 4: Rosen 47