Thus mathematics is not merely the science of measurement and number but any study consisting of symbols and precise rules of operation. The most basic mathematical objects seem to be sets. These rules help us understand and reason with statements such as which in simple english means there exists an integer that is not the sum of two squares. The rules of logic specify the meaning of mathematical statements. Proof, sets, and logic department of mathematics boise state.
Basic concepts of set theory, functions and relations. The author version from june 2009 corrections included. The mathematical analysis of logic maintained that the essential character of mathematics lies in its form rather than in its content. With a mathematical 777 mathematical logic gas logic. Also preliminaries from partee 1979, fundamentals of mathematics for linguistics. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. The concept of sets is an essential foundation for various other topics in mathematics. Many sets are given a shorthand notation in mathematics because they are used so frequently. At the hardware level the design of logic circuits to implement in. The converse of this statement is the related statement. The central concept of deductive logic is the concept of argument form.
Set, in mathematics and logic, any collection of objects elements, which may be mathematical e. A scientist or engineer needs more than just a facility for manipulating formulas and a. We call proofs arguments and you should be convincing the reader that what you write is correct. With a customers who viewed logic and set theory with applications also viewed logic and set theory. The emphasis here will be on logic as a working tool. Noted for the economy of its presentation, it includes a wealth of basic and key results from all parts of mathematical logic. Variables and connectives propositional logic is a formal mathematical system whose syntax is rigidly specified. In such a treatment the notion of set itself is left undefined, but the axioms about sets are suggested by thinking of a set as a collection of mathematical objects. An introduction to higher mathematics whitman people. Only that part of set theory is included that is closely related to mathematical logic. What are the best books on mathematical logic and set. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. Mathematical logic, also called logistic, symbolic logic, the algebra of logic, and, more recently, simply formal logic, is the set of logical theories elaborated in the course of the last nineteenth century with the aid of an artificial notation and a rigorously deductive method. The more you see your proofs in this light, the more enjoyable this course will be.
Logic also, it is never a real restriction to assume that distinct quanti. Set theory for computer science university of cambridge. Introduction to logic and set theory 202014 bgu math. The order of the elements in a set doesnt contribute anything new. A mathematical introduction to the theory and applications of logic and set theory with an emphasis on writing proofs. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. Its only about 100 pages and a lot of the details are left as exercises for the reader, but in my opion the. These have included hodges 1977, logic, hamilton 1978, logic for mathematicians, boolos and jeffrey 1980, computability and logic, scott et al. Cantors diagonal argument to show the reals form a uncountable set. Mathematics introduction to propositional logic set 1. Basic set theory a set is a many that allows itself to be thought of as a one. Each variable represents some proposition, such as you wanted it or you should have put a ring on it.
In mathematics, the terms and, or, not have precise meaning and are often written as symbols instead of words. However, this is not to suggest that logic is an empirical i. There are many fine books on mathematical logic, but mendelsons textbook remains a sure choice for a first course for its clear explanations and organization. In this introductory chapter we deal with the basics of formalizing such proofs. Set theory chiara ghidini fbkirst, trento, italy 20142015 chiara ghidini mathematical logic outline set theory relations functions 1 set theory basic concepts operations on sets operation properties 2 relations properties equivalence relation 3 functions. Also, in saying that logic is the science of reasoning, we do not mean. Boole founded not only a new algebra of sets but also a formal logic. The evolution of our number system can be summarized roughly as the series of set inclusions. Mathematical logic textbook thirdedition typeset and layout. Disjunction is related to the concept of union of two sets. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained.
A course in mathematical logic antonino salibra november 12, 2015 1 notations n. This is being written as a textbook for math 502, logic and set. The converse of this statement is the related statement if q, then p. Basic sets, comprehension, indexed sets, unions, intersections, products, disjoint unions, powersets. Members of a herd of animals, for example, could be matched with stones in a sack without members. Its first part, logic sets, and numbers, shows how mathematical logic is used to develop the number structures of classical mathematics. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic.
Propositional logic enables us to formally encode how the truth of various propositions influences the truth of other propositions. Mathematical logic on numbers, sets, structures, and. Excellent as a course text, the book presupposes only elementary background and can be used also for selfstudy by more ambitious students. The study of these topics is, in itself, a formidable task.
Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Some of the reasons to study logic are the following. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. This book, presented in two parts, offers a slow introduction to mathematical logic, and several basic concepts of model theory, such as firstorder definability, types, symmetries, and elementary extensions. Sets, logic and maths for computing david makinson springer. Mathematical logic and sets in this chapter we introduce symbolic logic and set theory. A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. Logic is the basis of all mathematical reasoning, and of all automated reasoning. If you have some experience in proof writing, i would say read naive set theory by paul halmos. Uniqueness, proving notall statements, the principles of mathematical induction. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science.
The order of the elements in a set doesnt contribute. Sets may be thought of as a mathematical way to represent collections or groups of objects. Like logic, the subject of sets is rich and interesting for its own sake. Social choice theory, in particular majority judgment and. This easytofollow textbook introduces the mathematical language, knowledge and problemsolving skills that undergraduates need to study computing. An introduction to elementary set theory guram bezhanishvili and eachan landreth. Basic mathematical notation and argument, including proof by contradiction, mathematical induction and its variants. Some sections of chapter 3 are partly descriptive, providing a perspective on. There are various symbolic logic systems, and indeed mathematical logic is its own branch of mathematics, but here we look at that portion of. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Determine if certain combinations of propositions are. For example, the statement if x 2, then x2 4 is true while its converse if x2. An introduction to set theory department of mathematics.
In fact, logic is a major and active area of mathematics. Starting with the basics of set theory, induction and computability, it covers. Sets and probability department of mathematics, texas a. We will develop some of the symbolic techniques required for computer logic.
Mathematical logic for computer science is a mathematics textbook, just as a. As we have seen from our examples, sets may contain a. The language is in part qualitative, with concepts such as set, relation, function and recursioninduction. The system we pick for the representation of proofs is gentzens natural deduction, from 8.
They are not guaranteed to be comprehensive of the material covered in the course. Fundamentals of mathematical logic logic is commonly known as the science of reasoning. Set operations, venn diagrams, set identities, subsets. A set is a collection of well defined and distinct objects. Highlighting the applications and notations of basic mathematical concepts within the framework of logic and set theory, a first course in mathematical logic and set theory introduces how logic is used to prepare and structure proofs and solve more complex problems. For those of you new to abstract mathematics elementary does not mean simple though much of the material. Mathematical logic co nstitutes the base ment of mathematics. Logic the main subject of mathematical logic is mathematical proof. In line with our introductory remarks on circularity we initially treat set theory naively, i. But mathematical logic also uses the concept of sets, e. Our objective is to reduce the process of mathematical reasoning, i. Rather, logic is a nonempirical science like mathematics.
Nb note bene it is almost never necessary in a mathematical proof to remember that a function is literally a set of ordered pairs. A first course in mathematical logic and set theory wiley. There are some sets or kinds of sets that hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the empty set, denoted or a set with exactly one element, x, is a unit set, or singleton, x. In mathematics, the notion of a set is a primitive notion. Two sets are equal if and only if they have the same elements. Instead, logic and mathematics provide a concise language as a means of expressing knowledge, which is something quite different from logic and mathematics. Learn and master all of discrete math logic, set theory, combinatorics, graph theory, etc 4. Sets and elements set theory is a basis of modern mathematics, and notions of set. Mark rated it really liked it sep 19, check out the top books of the year on our page best books of finite and denumerable sets hartogs theorem. Long chapter i called background material is rather similar to great chapter i on zfc set theory in kunens excellent 2009 book the foundations of mathematics logic s. Complex issues arise in set theory more than any other area of pure mathematics. We often deal with groups or collection of objects in real life, such a set of books, a group of students, a list of states in a country, a collection of baseball cards, etc. We will return to sets as an object of study in chapters 4 and 5.
Propositional logic propositional logic is a mathematical system for reasoning about propositions and how they relate to one another. Formulae are built inductively using logical operators and quantifying over individual variables. Logic, proofs, and sets jwr tuesday august 29, 2000 1 logic a statement of form if p, then q means that q is true whenever p is true. Relation between compound statements and sets in set theory. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. For our purposes, it will sufce to approach basic logical concepts informally. A mathematical model that we will use often is that of. Mathematical logic hannes leitgeb october 2006 these lecture notes follow closely. Fv is used for the set of free variables of an expression. Solomon feferman, stanford university, january 2001 the book remains an excellent introduction to logic. Validity, entailment, and equivalence of boolean propositions.
Duplicates dont contribute anythi ng new to a set, so remove them. We will need only a few facts about sets and techniques for dealing with them, which we set out in this section and the next. An argument is a sequence of statements aimed at demonstrating the truth of an assertion a claim. A first course in mathematical logic and set theory.
The intuitive idea of a set is probably even older than that of number. Jul 12, 2018 list of books of logic and set theory. Although the necessary logic is presented in this book, it would be bene. It will cover truth tables, logical equivalences, validity and invalidity, soundness, sets, set operations, venn. A statement and its converse do not have the same meaning. The next chapters are consecutively about propositional logic, sets finite and infinite, predicate logic, arithmetic and godels incompleteness theorems, modal logic, philosophy of language, intuitionism and intuitionistic logic, applications prolog. Set symbols of set theory and probability with name and definition. Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Every statement in propositional logic consists of propositional variables combined via logical connectives. This is a systematic and wellpaced introduction to mathematical logic. The language of set theory can be used to define nearly all mathematical. This book by master expositor kenneth kunen, emeritus at university of wisconsinmadison, is a newly rewritten 2011 update of his well regarded, rather standard 1980 edition, still available at set theory an introduction to independence proofs studies in logic and the foundations of.
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