Lecture Notes: Discrete Mathematics

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1 Lecture Notes: Discrete Mathematics GMU Math Sring 2007 Alexei V Samsonovich Any theoretical consideration, no matter how fundamental it is, inevitably relies on key rimary notions that are acceted without definition. In our case, examles of notions that we shall accet without definitions are: an object, a value, a concet, a symbol, to name just a few. In fact, most English words used here are not defined yet exected to be understood. Given this agreement, our first ste is to give useful definitions of the key terms, notations and concets wisely selected among others. Below in this text, whenever a key term is introduced, it is tyed in bold italic. Synonyms are listed in arentheses. An examle follows immediately. Discrete mathematics (sometimes understood as finite mathematics) is the study of mathematical structures that do not suort or reuire the notion of continuity. In articular, all comuter imlementations of continuous mathematical structures involve their finite discretization in a certain sense. Toics of discrete mathematics include logic, sets, number theory, combinatorics, grahs, algorithms, robability, information, comlexity, comutability, and more. Lecture notes included here should be considered comlementary rather than alternative to the textbook. The intent is to hel with understanding of difficult concets: sometimes by taking the reader to a higher level of abstraction that oens a better view of the main subject, and sometimes by sueezing the essence from long ages into just a few lines. Lecture 1: Logic It is generally understood that mathematics lays the role of a theoretical fundament for all modern natural sciences. In the same sense, logic can be viewed as a theoretical fundament for mathematics in general and for discrete mathematics in articular. It is not surrising therefore that we start our consideration with logic. In order to introduce concets of logic from first rinciles, we need to introduce some other general notions first. By introduce I do not mean rigorously define. Many of these notions will be addressed in more detail and used extensively in subseuent lectures, so in a sense the following three aragrahs are just an outline. A set is a collection of distinct objects called elements (or members) of the set. In articular, the notion of a set does not allow for multile instances (reetitions) of the same element in the set. In contrast, a seuence is an ordered collection that allows for reetition of its elements. An ordered n-tule is a seuence of n elements. If X is a set, then X n (the n-th ower of X) is the set of all ordered n-tules over X ( over X here means comosed from elements of X). An n-lace relation on X is a subset of X n.

2 Sulementary Lecture Notes for Math Alexei V. Samsonovich 1/27/ An exression is a finite seuence of elements-symbols (the entire set of symbols of a given language is called alhabet or vocabulary). A well-formed exression satisfies certain grammar rules (that need to be secified searately in each articular case of a formal theory). Below the term exression will refer to well-formed exressions only. A language is a set of exressions over an alhabet. A formal axiomatic theory is a language in which a subset of exressions are called axioms, and the rest are connected to the axioms via relations called inference rules. A seuence of these connections that starts from axioms is called a roof, and the exression at the end of this seuence is a theorem. A theory is decidable, if there is an algorithm to determine for any given exression whether it has a roof. Here we comare two theories: roositional calculus and redicate calculus (both are addressed in more detail below). Elements of roositional calculus are truth-valued (Boolean) variables, constants and exressions constructed from them using connectives (AND, OR, etc.). Elements of redicate calculus are variables that may take values of any nature (e.g., numbers, eole), lus arbitrary truth-valued functions of those variables called redicates, lus connectives and uantifiers used to construct exressions. A variable is a symbol that can be associated with (bound to) a value, an object, a set, or an exression. A variable can be free or bound. Variables may have tyes: restrictions on the nature of objects to which they can bind. On the other hand, a uantifier secifies a set of objects to which a variable is (to be) bound. While it is ossible to design an unlimited number of uantifiers, two of them are commonly used in most cases: the universal uantifier, secifying that something alies (the variable can be bound) to any element of a certain category, and the existential uantifier, secifying that something alies to at least one element. Given the above outline, we can define roositional calculus (roositional logic) as a formal axiomatic theory L with the alhabet, syntax, set of axioms and inference rules that follow below. The alhabet includes connectives (,,,, ), arentheses: (, ), and variables denoted here by lowercase English letters that may aear with or without subscrits. Each variable in L is bound to a roosition (a statement), which is an exression that is either true or false (i.e., has a definite truth value). The set of exressions in L can be formally given as follows. 1. All letters that denote statements with definite truth values are exressions of L. 2. If B and G are exressions of L, then so are ( B) and (B G). 3. If B, G and D are laceholders for exressions of L, then the following axiom schemas define axioms of L: a. (B (G B)) b. ((B (G D)) ((B G) (B D))) c. ((( G) ( B) ((( G) B) G)) 4. It suffices for L to have the only inference rule, modus onens: ((a b, a) b. Again, in summary, roositional calculus (roositional logic) can be defined as a formal axiomatic theory in which the alhabet includes variables denoted by letters (e.g.,

3 Sulementary Lecture Notes for Math Alexei V. Samsonovich 1/27/2007 3, 1 ), symbols 1 and 0 used for tautology and contradiction, connectives (,,,, ) and arentheses: (, ), the set of axioms includes those given by the above schemas (a), (b) and (c), and the only inference rule is modus onens. We can avoid using unnecessary arentheses, if we agree on a articular order of recedence for connectives, e.g.:,,,,, and then and other symbols of meta-language. We can reason about variables in L without knowing their values, yet we assume that they have definite values within the set {T, F}. This feature searates roositional calculus from redicate calculus, in which redicates may have free variables. In L, all exressions are roositions. A direct analogy with regular numerical exressions can be seen. Indeed, roositional logic is isomorhic to binary arithmetic (calculus of Boolean numbers: zeros and ones), as the following translation table shows. This translation rovides an intuition into L and can be taken as a art of its definition. Table 1. L is isomorhic to Boolean arithmetic. Proositional logic Boolean arithmetic T 1 F (XOR) (1-) 0 Given this table, you can confidently solve your HW1 roblems (or you can use the truth tables), yet there is a more convenient way of doing this: by using euivalence relations listed on age 24 in the textbook. At least, with this table you should have no roblem understanding what L is about: it is a calculus in which all variables and exressions take only two values. In this sense it is much simler than redicate calculus. Therefore, for examle, it is not surrising that and cannot be simultaneously false in roositional logic. It is also clear that there may be cases when P Q holds and P Q does not (generally, whenever P Q holds, P Q is a tautology, and vice versa). The analogy between and multilication, and addition with regard to commutativity, associativity and distributivity is straightforward. In articular, it imlies that euivalence relations allow us to reduce any logical exression to its disjunctive normal form (dns): i.e., a logical sum (disjunction) of disjuncts, each of which is either a logical roduct (conjunction) of literals (i.e., variable letters or their negations) or a single literal. The counterart of dns is a conjunctive normal form (cns) defined similarly, with swaing

4 Sulementary Lecture Notes for Math Alexei V. Samsonovich 1/27/ and. These forms can be useful when you want to comare two exressions. A dns / cns is called full, if each of its disjuncts / conjuncts is a minterm (contains each variable used in the form exactly once). Every non-contradiction / non-tautology can be reduced to a full dns / cns. An exression is satisfiable if it is true for some values of its variables. A logical argument is a finite set of roositions called remises or hyotheses followed by a roosition called conclusion. In a valid argument, the conclusion is true whenever remises are true. A sound argument is a valid argument in which remises are true. An arguments can be written in a column form (below), or as a comound statement with an imlication connecting remises to the conclusion. As a comound statement, a valid argument is a tautology. Moreover, its variables can be treated as laceholders for exressions (substitution theorem). All this is useful for roblem solving. A table of useful arguments some inference rules derived from modus onens is given below. Table 2. Inference rules. Modus onens Modus tollens introduction Contraositive Case analysis r r r Vacuous roof: introduction elimination introduction elimination Disjunctive syllogism Resolution r r Chain rule r r Contradiction F

5 Sulementary Lecture Notes for Math Alexei V. Samsonovich 1/27/ First-order logic (FOL, first-order redicate calculus) is an extension of roositional logic L that includes uantifiers, redicates (see above) and variables with arbitrary values beyond {T, F}, with the excetion for redicates themselves, which makes this theory first-order (without this restriction we run into logical aradoxes). The set of axioms and the set of rules of inference in FOL, as comared to L, is augmented with (intuitively obvious) axioms and arguments involving uantifiers. A higher-order logic differs from FOL in that its redicates are allowed to have redicates as their arguments. The notion of logic, esecially in comuter science, is raidly exanding in many dimensions, including fuzzy logic, many-valued logic, uantum logic, modal logics (temoral, dynamic, deontic, alethic, eistemic, etc.). Related calculi include situation calculus, fluent calculus, event calculus, etc. Further Reading: Hamburger, H., and Richards, D. (2002) Logic and Language Models for Comuter Science. Uer Saddle River, NJ: Prentice Hall. Mendelson, E. (1997) Initroduction to Mathematical Logic. New York: Chaman & Hall.

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