Set theory is also the most philosophical of all disciplines in mathematics. An overview of logic, proofs, set theory, and functions. That will be unifying theme throughout the entire quarter, and youll see this come up in the rest of this handout. What are the best books on mathematical logic and set theory.
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. While we usually list the members of a set in a standard order if one is available there is no. Although no one actually uses this term, i like to call it the elemental theory of sets. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. The set t 2,3,1 is equal to s because they have the same members. By 1900, set theory was recognized as a distinct branch of mathematics. Elementary set theory proof techniques download ebook. Proofs of settheoretic relations and equalities among sets are among the simplest types of proofs and therefore present an excellent opportunity to familiarize. This alone assures the subject of a place prominent in human culture. Questions are bound to come up in any set theory course that cannot be answered mathematically, for example with a formal proof.
If you have some experience in proof writing, i would say read naive set theory by paul halmos. At just that time, however, several contradictions in socalled naive set theory were discovered. Algebraic proofs of sets set theory mustafa jarrar. Its only about 100 pages and a lot of the details are left as exercises for the reader, but in my opion the. Logical statements be combined to form new logical statements as follows. An introduction to writing proofs, the basic types of proofs, and an introduction to important mathematical objects such as functions and relations. A logical statement is a mathematical statement that is either true or false. Introduction to mathematical arguments background handout for courses requiring proofs by michael hutchings a mathematical proof is an argument which convinces other people that something is true. The notion of set is taken as undefined, primitive, or basic, so we dont try to define what a set is. These will be the only primitive concepts in our system. There are other conceptions of set, but although they have genuine mathematical interest they are not our concern here. Basic concepts of set theory, functions and relations.
On the platonic level, this is intended to communicate something about proof, sets, and logic. Click download or read online button to get elementary set theory proof techniques book now. Set theory frank stephan semester i, academic year 20092010 set theory deals with the fundamental concepts of sets and functions used everywhere in mathematics. View notes endertonsolutions from math 361 at rutgers university. The notions of set theory, and the ideas behind many of the proofs, present themselves to the inner eye in vivid detail. We derive them by going back to the definitions of intersection, union, universal set and empty set, and by considering whether a given element is in, or not in, one or more sets. Proof theory was created early in the 20th century by david hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics in arithmetic number theory, analysis and set theory.
Set theory has its own notations and symbols that can seem unusual for many. Cantor initiated the study of set theory with his investigations on the cardinality of sets of real numbers. Find materials for this course in the pages linked along the left. How to write proofs in set theory with the math sorcerer 4. In earlier versions of our logic style manual, we tended to omit these. This is being written as a textbook for math 502, logic and set theory, and math 522, advanced set theory, at boise state university, on the practical level. Find, with proof, identities for the operations set union and set intersection. How to write proofs in set theory with the math sorcerer. Today, when mathematicians talk about set theory as a field, they usually mean axiomatic set theory.
If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools. Proofs of graphtheoretic results and methods are usually not given in a completely rigorous. The amount of detail that an author supplies in a proof. Elementary set theory proofs mathematics stack exchange.
The intersection of sets a and b is the set a\b fx. The foundational rules of set theory the laws listed below can be described as the foundational rules of set theory. A proof is an argument intended to convince the reader that a general principle is true in all situations. We give a proof of one of the distributive laws, and leave the rest for homework. Famous theorems of mathematicsset theory wikibooks. Already in his famous \mathematical problems of 1900 hilbert, 1900 he raised, as the second. Here we perform a couple containment proofs from naive set theory. Set operations and the laws of set theory the union of sets a and b is the set ab fx. Mcgeoch amherst college 1 logic logical statements. The basis of graph theory is in combinatorics, and the role of graphics is only in visual. 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. The axioms of set theory of my title are the axioms of zermelofraenkel set theory, usually thought ofas arisingfromthe endeavourtoaxiomatise the cumulative hierarchy concept of set. When expressed in a mathematical context, the word statement is viewed in a.
Proof, sets, and logic department of mathematics boise state. The logic and set theory are presented in a naive way. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Here we denote logical statements with capital letters a.
In order to eliminate such problems, an axiomatic basis was developed for the theory of sets analogous to. Discrete mathematicsset theorypage 2 wikibooks, open. Set theory empty set the set that contains no element is called the empty set or null set. Here is another set equality proof from class about set operations. 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. Nb note bene it is almost never necessary in a mathematical proof to remember that a function is literally a set of ordered pairs. Chapter 4 set theory nanyang technological university. Mat231 transition to higher math proofs involving sets fall 2014 4 11. That is, it is possible to determine if an object is to be included in the set. Math isnt a court of law, so a preponderance of the evidence or beyond any reasonable doubt isnt good enough. 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.
Chapter 1 logic and set theory to criticize mathematics for its abstraction is to miss the point entirely. Proof involving sets and indexed families of sets by. All sets are made of elements, and theyre completely defined by their elements. But even more, set theory is the milieu in which mathematics takes place today.