Exercises in set theory pdf

Set theory deals with the fundamental concepts of sets and functions used everywhere in mathematics. Because the fundamentals of set theory are known to all mathematicians, basic problems in the subject seem elementary. Rosen uses the latter, but the former is actually more standard. We can specify a set by listing the elements within braces. These notes for a graduate course in set theory are on their way to be. The standard form of axiomatic set theory is the zermelofraenkel set theory, together with the axiom of choice. Set theory is likely to be around long after most presentday programming languages have faded from memory. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. This means that 1,2,3 is a set but 1,1,3 is not because 1 appears twice in the second collection. The importance of model theory lies in the observation that mathematical objects can be cast as models for a language. Chapter 1 introduction in this course, we will develop set theory like any other mathematical theory on the basis of a few given axioms and generally accepted practices of logic. Thus, ais the set of all real numbers whose square is one. In the second part i present the topics and techniques that i believe every set. This accessible approach to set theory for upperlevel undergraduates poses rigorous but simple arguments.

For chapter 24, this is not too helpful since reading shelahs book on pcf theory is not for the light hearted. Set theory is also the most philosophical of all disciplines in mathematics. We provide a few exercises which allow us to practice notions related to power sets. Then by the axiom schema of comprehension, there is a set bdf x2vw g. Give an example of a semigroup without an identity element. This chapter introduces set theory, mathematical in.

The purpose of this module is to introduce language for talking about sets, and some. When expressed in a mathematical context, the word statement is viewed in a. The reader might contemplate the examples before reading the solutions, but this is not mandatory. However, the subcollection of books we have read is much smaller. Formal set notation description informal english description 2, 4, 6, 8, 10, the set of all positive even integers, 3, 1, 1, 3, the set of all odd integers n n 2m for some y. For any two sets x and y, either there is a onetoone function from. In this case, we say ais a subset of bor equivalently that ais contained in b. Prove the associative law for intersection law \2\prime\ with a venn diagram. We can express the fact that 4 is not an element of s by writing 4. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is, but we can give an informal description, describe. Moreover, the theory of inner models has emerged as a major part of the large cardinal theory. Formal set notation description informal english description a 2, 4, 6, 8, 10, the set of all positive even integers b, 3, 1, 1, 3, the set of all odd integers c n n 2m for some y the set of all positive even integers using the convention that 0 is not a natural number.

In each of the following cases, suppose we are given sets b k for each k2n. Here are three simple statements about sets and functions. A small number of exercises however will require some knowledge of point set topology or of set theoretic concepts such as cardinals and ordinals. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Axioms and set theory home mathematics university of waterloo. Many different systems of axioms have been proposed. Prove the idempotent law law 6 using basic definitions. We know that the power set is the set of all subsets. They look like they could appear on a homework assignment in an undergraduate course. A set is a collection of objects, called elements of the set. Sets and venn diagrams australian mathematical sciences.

Set theory \a set is a many that allows itself to be thought of as a one. An introduction to set theory university of toronto. Sets and functions are among the most fundamental objects in mathematics. Amattmat 118 notes and exercises on sets and functions 1.

This pdf version is the result of a first pass through the text in november 2012 with the aim of preparing an official online second edition. Illustrate each of the following by shading the venn diagrams. Empty set subset properties theorem s empty set is a subset of any set. Formal set notation description informal english description 2, 4, 6, 8, 10, the set of all positive even integers, 3, 1, 1, 3, the set. The examples and exercises in this section give some applications of this. Exercises exercises on the properties of relations optional 1. What is the set of colors appearing on both the american. It is difficult to give an account of the axiomatic theory of sets at an elementary level, which is why, intuitively, we shall define a set as a collection of objects, named elements or points of the set. Basic concepts of set theory, functions and relations. I have permission from my publisher to post a revised version of my book elementary set theory with a universal set which has gone out of print online.

The examples in this module have shown how useful sets and venn diagrams are in co. Subsets a set a is a subset of a set b iff every element of a is also an element of b. Give a verbal description of the set 1,4,8,9,16,25. Show that the set of all xsuch that x2aand xbexists. Then by the axiom schema of comprehension, we know that such a set does exist. A formal treatment of set theory was first undertaken at the end of the 19th century and was finally codified in the form of the. It is a foundational tool in mathematics the idea of grouping objects is really useful examples. One set may be involved as a universe with many di erent interpretation functions of the language l. Some naive set theory, developing ideas about ordinals, is occasionally useful, and the abstraction of this setting makes the set theory seem less farfetched or ba.

Set theory frank stephan semester i, academic year 20092010 set theory deals with the fundamental concepts of sets and functions used everywhere in mathematics. Chapter 2 the zermelo fraenkel axioms of set theory the axiom of extensionality. It is unfortunately true that careless use of set theory can lead to contradictions. The proof of the following theorem is left as an exercise. These will be the only primitive concepts in our system. The language of set theory has only one extralogical symbol,1 the binary relationsymbol ifmisasetand eisabinaryrelationon m. Set theory basic building block for types of objects in discrete mathematics. Cantor initiated the study of set theory with his investigations on the cardinality of sets of real numbers. X meaning x belongs to the set x, or x is an element of x tipically we use capital letters to designate sets and small letters to designate elements of a set. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. A relation is an equivalence if it is reflexive, transitive and symmetric. What is the set of colors appearing on both the american flag and the ja. Here is my submitted draft 2018 on representation of functions in third order logic this paper has appeared. Some naive set theory, developing ideas about ordinals, is occasionally useful, and the.

The strong tradition, universality and neutrality of set theory make it. The set of all cities with population more than one million people. To prove that a given set ais contained in b, one needs to show that x2ax2b. In the following sentences, identify the property, and translate the sentence to set notation. Amattmat 118 notes and exercises on sets and functions. The subcollection of phone numbers of our personal contacts is much more manageable. If a is a set and x is an element of a, we write x 2 a. The second primitive notion of set theory is the notion of belonging. The resulting axiomatic set theory became known as zermelofraenkel zf set theory. If gis a nonempty set, a binary operation on g is a function. You have to hand in an exercise marked with a star in weeks 3 to 6, weeks 7 to 9.

Issues about data structures used to represent sets and the computational cost of set operations. Each of the axioms included in this the ory expresses a property of sets that is widely accepted by mathematicians. If every element of the set a is an element of the set b and every element of the set b is an element of the set a, then a b. Some additional exercises are given, which are variations on the worked examples. Set theory and topology at its essence, all of mathematics is built on set theory. The set of all phone numbers is too large for most of us to handle. Sep 30, 2020 exercises in set theory by laurence e. In this chapter, we will introduce some of the basics of sets and their properties. The operation is often called multiplication and if x. Let ibe a nonempty set, let fa ig i2ibe a family of sets indexed by. In an advanced class on set theory, you would see a more careful description of a set, which avoids this fuzzy use of the term elements. Describe the following sets in both formal and informal ways.

We then say that the set m is included in the set n. Unpack each of the following sets and see if you can. Joshua cooper, fall 2010 determine which of the following statements are true and which are false, and prove your answer. Formal set notation description informal english description a 2, 4, 6, 8, 10, the set of all positive even integers b, 3, 1, 1, 3. Set theory is indivisible from logic where computer science has its roots. Replace the axiom of existence by the following weaker postulate. Set theory for computer science university of cambridge.

Which of the following is sets is shown with roster notation. The examples in this module have shown how useful sets and venn diagrams are in counting. Exercises for unit i general considerations ucr math. A set theory textbook can cover a vast amount of material depending on the mathematical. A revised and corrected republication of set theory, originally published in 1971 by addisonwesley publishing company, reading, massachusetts.

We must show the following implication holds for any s x x x s since the empty set does not contain any element, x is. The universal set, the set that contains all the sets examined further, which has the containing elements nature one and the same, will be denoted by. In the exercises that follow it is most important that you outline the logical procedures or methods you use. The symbol n has the same meaning as in the context of set theory. Sets a set is a collection of things called elements. Cantor initiated the study of set theory with his investigations. Problems are sorted into different sets based on how hard they are to solve.

Which of the following are true and which are false. Ion goian raisa grigor vasile marin florentin smarandache. A set can be represented by listing its elements between braces. A set is a collection of objects satisfying a certain set. The group b questions require a deeper understanding. Note that in the second identity, we show the number of elements in each set by the corresponding shaded area. Is it ever possible for one set to be both an element and a subset of another.

Complex issues arise in set theory more than any other area of pure mathematics. Because the set a a, e, i, o, u contains 5 elements. A large number of exercises are interspersed throughout the text, and it is intended that the reader perform a signi cant fraction of these exercises while going through the text. So the set containing the first 10 letters in the english alphabet would. The model is both the universe and the interpretation function. With we will note the set of all parts of set, in short. Vowels in the english alphabet v a, e, i, o, u first seven prime numbers. Most exercises in jech are definitely on topic here though there is a useful trick which is to dig through the papers in the historical notes until a solution is found. It has nice set of exercise problems but i can promise you that naive set theory by halmos will still be as painful to read for a beginner in set theory.

Although elementary set theory is wellknown and straightforward, the modern subject, axiomatic set theory, is both conceptually more di. 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. Group theory notes michigan technological university. In an advanced class on set theory, you would see a more.