Logic and set theory examples

At the end of the course, students will be able to 1 detect the logical structure behind simple puzzles 2 be able to manipulate logical expressions 3 explain the connection between logic and set theory 4 explain the differences between natural, integer, rational, real and complex numbers 5 recognise different basic proof techniques. Proof theory is, quite logically, the study of formal proofs. The set of rational numbers is a proper subset of the set of real. Set theory definition is a branch of mathematics or of symbolic logic that deals with the nature and relations of sets. Set theory and logic supplementary materials math 103. Axiomatic set theory, which we will discuss in a later chapter, provides a foundation for. Ellermeyer august 18, 2000 set theory membership a setis a welldefined collection of objects.

The study of these topics is, in itself, a formidable task. Lecture 7 software engineering 2 propositional logic the simplest, and most abstract logic we can study is called propositional logic. This unit also covers subsets, the universal set, set builder notation, complement, intersection and union. What is the use of logic and set theory in computer. Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. Logic and set theory a rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. An introduction to elementary set theory guram bezhanishvili and eachan landreth 1 introduction in this project we will learn elementary set theory from the original historical sources by two key gures in the development of set theory, georg cantor 18451918 and richard dedekind 18311916. Then the basic operations of addition, subtraction, multiplication, and division of binary number can actually be accomplished using the right combination of gates. Leader, lentterm 2005, 2010 chapter 1 propositional logic 1 chapter 2 wellorderings and ordinals 7 chapter 3 posets and zorns lemma 16 chapter 4 predicate logic 24 chapter 5 set theory 34 chapter 6 cardinals 43 bonus lecture incompleteness examples sheets prerequisites.

For example, a deck of cards, every student enrolled in. Questions are bound to come up in any set theory course that cannot be answered mathematically, for example with a formal proof. Fuzzy logic classical set theory a set is an unordered collection of different elements. Set theory tutorial problems, formulas, examples mba. Motivation when you start reading these notes, the first thing you should be asking yourselves is what is set theory and why is it relevant. Use wolframalpha to visualize, compute and transform logical. Basic definitions and notation, types of sets, equality, and venn diagrams are presented. Obviously, all programming languages use boolean logic values are true and false, operators are and, or, not, exclusive or. A solid foundation on sets is provided for students of all ages. To indicate that an object x is a member of a set a one writes x. Traditionally, the and operation is written like multiplication.

A set may be defined by a membership rule formula or by listing its. Such as, sets could be a collection of odd numbers, even numbers, natural numbers, whole numbers, real or complex numbers and all the set of numbers which comes in the number line. Predicate logic predicate logic, set theory, and functions. The big questions cannot be dodged, and students will not brook a flippant or easy answer. Set theory is indivisible from logic where computer science has its roots. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Set theory for computer science university of cambridge. Mar 17, 2020 fuzzy logic has been applied to various fields, from control theory to ai. State the axiom of foundation and the principle of induction, and show that they are equivalent in the presence of the other axioms of zfc. In natural language, the coordinating conjunction and.

Use features like bookmarks, note taking and highlighting while reading set theory and logic dover books on mathematics. For example, suppose that committee a, consisting of the 5 members jones, blanshard, nelson, smith, and hixon. The language of set theory can be used to define nearly all mathematical. This instructional unit on sets provide a stepbystep introduction to sets and set theory. Examples like this helped make set theory a mathematical subject in its own right. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. A u bc, ac intersection bc, a intersection bc, ac u bctags. In naive set theory, a set is a collection of objects called members or elements that is regarded as being a single object. If the object x is a member of the set a, then we write x a which is read as x is a member of aorx belongstoaorx is in aorx is an. Set theory as a programming language, logic proof verifiers, topolgy. This resource guide discusses program theory and logic models. For example, the expression there are no classes at texas. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p.

In mathematical logic, a firstorder theory is given by a set of axioms in some language. State the compactness theorem and the decidability theorem, and deduce them from the completeness theorem. Set theory, branch of mathematics that deals with the properties of welldefined collections of objects, which may or may not be of a mathematical nature, such as numbers or functions. In this module weve seen how logic and valid arguments can be formalized using mathematical notation and a few basic rules. Let a, b, c be three sets as shown in the following venn diagram. A set s of propositions is called nitary if there exists a nite set t of propositions. As we have seen from our examples, sets may contain a.

It will cover truth tables, logical equivalences, validity and invalidity, soundness, sets, set. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality the concept that corresponds to size, that is, the number of elements, of a finite set as the whole. Set theory and logic dover books on mathematics reprint. For example, suppose your friend promises if it is sunny tomorrow, i will ride my bike. For those of you new to abstract mathematics elementary does not mean simple though much of the material. In mathematics, the notion of a set is a primitive notion. In this tutorial, we look at some solved examples to understand how set theory works and the kind of problems it can be used to solve. Department of pure mathematics and mathematical statistics, university of cambridge. We can list each element or member of a set inside curly brackets like this. They are not guaranteed to be comprehensive of the material covered in the course. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. Basic set theory lx 502 semantics i september 11, 2008 1. The term logical conjunction is also used for the greatest lower bound in lattice theory.

Though propositional logic will prove a useful tool to describe certain aspects of meaning, like the reasoning in 1, it is a blunt. If the object x is a member of the set a, then we write x a which is read as x is a. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Georg cantor in the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains in predicate logic form sets as well. For example, an and gate has two input wires a, b and one output c. Discrete mathematics, set theory, and logic discrete math, graph theory, combinatorics, recurrence relations difference equations, set theory, gene. It was designed to allow the computer to determine the distinctions among data which is neither true nor false. Additional materials on logic contributed by eugenio omodeo and domenico cantone. Almost everyone knows the game of tictactoe, in which players mark xs and os on a threebythree grid until one player makes three in a row, or the grid gets filled up with no winner a draw. Set theory has its own notations and symbols that can seem unusual for many. This entry lists some of the more common examples used in model theory and some of their properties.

Using the same example, if a is the set of all teams that took part in the 2006 football world cup and b is the set of teams that reached the quarterfinals, then b is said to be a. While im unsure any straight uses of set theory, fuzzy set theory gets used directly in quite a few areas engineering, medicine, business, social sciences where information is incomplete. For example, a deck of cards, every student enrolled in math 103, the collection of all even integers, these are all examples of sets of things. If you apply t he axiom of replacement to a functionclass f, you must explain clearly why f is indeed a functionclass. Beyond that, set theory introduces the concept of sets containers of elements and provides a number of operators, inclu. Introduction to logic and set theory 202014 bgu math. What is the use of logic and set theory in computer science.

Mathematics introduction to propositional logic set 1. From wikibooks, open books for an open world logic. Symbolic logic and set theory are intertwined and lie at the foundations of mathematics. Set theory is also the most philosophical of all disciplines in mathematics. Chapter 4 set theory \a set is a many that allows itself to be thought of as a one. Website devoted to highfalutin intellectual issues. Any object which is in a set is called a member of the set. We will call this a true statement if they keep their promise. Set of natural numbers 1,2,3, set of whole numbers 0,1,2,3, each object is called an element of the set. For each of the following sets, draw a venn diagram and shade the area representing the given set. Department of pure mathematics and mathematical statistics. Introduction to propositional logic set 2 this article is contributed by chirag manwani.

In fact when george boole 18151864 first developed symbolic logic or boolean logic, he had the idea that his system could be used by lawyers, philosophers, and mathematicians alike to help put convoluted arguments on a firmer footing. Though aimed at a nonmathematical audience in particular, students of philosophy and computer science, it is rigorous. Oct 17, 20 modern set theory is it a religious belief system. Each set of examples is accompanied by a brief write up that highlights distinctive characteristics of the logic model and theory of change for that type of mentoring program, as well as key supporting research and additional resources. At the end of this section, there are a number of examples and worksheets that can help you develop a program theory and logic model.

Applications in the social sciences or applications of fuzzy logic in bioinformatics. Let us see the different types of symbols used in mathematics set theory with its meaning and examples. Logic and set theory to motivate this denition, one can think of p. Complex issues arise in set theory more than any other area of pure mathematics. However, most of the time, we only have an intuitive picture of what set theory should look like there are sets, we can take intersections, unions, intersections and subsets. Moreover, numerical values can be represented by a string of 1s and 0s in what we call binary notation. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. If there are a finite number of elements in a set, or if the elements can be arranged in a sequence, we often indicate the set simply by listing its elements. Bread a union b or the union of a and bis defined as the set that consists of all elements belonging to either set a or set b or both. Set symbols of set theory and probability with name and definition. Set b is said to be a subset of another set a when all elements of set b are also elements of set a.

For example, we can consider all the following sets of natural numbers. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. About the open logic project the open logic text is an opensource, collaborative textbook of formal meta logic and formal methods, starting at an intermediate level i. If you like geeksforgeeks and would like to contribute, you can also write an article using contribute.

Note that in the second identity, we show the number of elements in each set by the corresponding shaded area. Set theory and logic why understand set theory and logic applications. Find solutions for boolean algebra, set theory, and transfinite number problems. Mathematical logic is often used in proof theory, set theory, model theory, and recursion theory. Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. Get answers to your logic and set theory questions with interactive calculators. Together, mathematical logic and set theory have produced some of the most beautiful theorems in all of mathematics. Electricity will flow at c if and only if both a and b have current.

Basic set theory a set is a many that allows itself to be thought of as a one. An elementary introduction to logic and set theory. Set theory and logic is the result of a course of lectures for advanced undergraduates, developed at oberlin college for the purpose of introducing students to the conceptual foundations of mathematics. Set theory definition of set theory by merriamwebster. In programming languages, the shortcircuit and control structure. It will cover truth tables, logical equivalences, validity and invalidity, soundness, sets, set operations, venn. Logic and set theory virginia commonwealth university.

Something similar to the process of human reasoning. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Set theory, branch of mathematics that deals with the properties of welldefined collections of objects such as numbers or functions. Set theory and logic dover books on mathematics kindle edition by stoll, robert r download it once and read it on your kindle device, pc, phones or tablets. Program theory explains why a program is expected to work and a logic model illustrates a program theory. It can be written explicitly by listing its elements using the set bracket. The second is presumably the set of all positive odd numbers. Discrete mathematicsset theoryexercises wikibooks, open.