Basic concepts of set theory, functions and relations. In naive set theory, a set is a collection of objects called members or elements that is regarded as being a single object. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. Examples of discrete structures built with the help of sets. B x x a and x b this is the intersection of a and b. Because the fundamentals of set theory are known to all mathematicians, basic problems in the subject seem elementary. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. Set union the union of two sets a and b, written a.
Pure set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets. Sets are unordered and we do not distinguish 1 a,b. The language of probability the mathematics of probability is expressed most naturally in terms of sets. Thus, two sets are equal if and only if they have exactly the same elements. U x is an even number the set of all x in u such that x is an even number the set u is called the universe. Basic set theory a set is a many that allows itself to be thought of as a one. They look like they could appear on a homework assignment in an undergraduate course. A complement of a set let u be the universal set and a a subset of u. If there is a onetoone function from x into y and also a onetoone function from y into x, then there is a onetoone function from x onto y. Learn the basic concepts and formulas of set theory with the help of solved examples. We can use these sets understand relationships between groups, and to analyze survey data. B for example, if s is the set of all juices in the supermarket, and t is the set of all. The notation and symbols for sets are based on the operations performed on them.
The following notation is used to show set membership means that x is a member of the set a means that x is not a member of the set a. Two sets are disjoint if they have no elements in common, that is, a and b are disjoint if a. Note that there is no problem with the fact that di. It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. Discrete mathematicsset theoryexercises wikibooks, open. We write \a\in a\ to indicate that the object \a\ is an. Basically, the definition states it is a collection of elements. They are not guaranteed to be comprehensive of the material covered in the course. Sets are welldetermined collections that are completely characterized by their elements.
The inversion of a pitch class is the pitch class that is the same number of half steps away from c. In set theory, inversion means the same number of half steps in the opposite direction. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. B the formal definition presupposes a and b are sets. In set theory the term universal set is often used in place of universe of discourse for a given predicate. Set theory basic concepts of set theory hitbullseye. Two sets are equal if and only if they have the same elements, i. Three important binary set operations are the union u, intersection. These elements could be numbers, alphabets, variables, etc.
Sets we discussed in class how to formally show that one set is a subset of another and how to show two sets are equal. Definition cartesian product if a and b are sets, the cartesian product of a and b. A set may be defined by a membership rule formula or by listing its. A subset of the cartesian product a x b is called a relation from the set a to the set b. In set theory and, usually, in other parts of mathematics, a cartesian product is a mathematical operation that returns a set or product set or simply product from multiple sets. A \ b denotes the set difference of a and b and consists of the elements in a that. Enderton elements of set theory, academic press, 1977. To generalize this to arbitrary sets aand bwe rst need the concept of an ordered pair. A the set of all even numbers b 2, 4, 6 c 2, 3, 4, 6 here b. In mathematics, the notion of a set is a primitive notion. It is designed for a onesemester course in set theory at the advanced undergraduate or beginning. This alone assures the subject of a place prominent in human culture. We write x2ato mean that xis an element of a set a, we also say that xbelongs to aor that xis in a. A to indicate that the object a is an element, or a member, of the set a.
In maths, set theory was developed to explain about collections of objects. The accompanying latex tutorial will guide you through the process of installing it on your computer, and show you the basics of how to prepare a document in. In a similar manner, there are several ways to create new sets from sets that have already been defined. The difference of a and b, denoted by a b, is the set containing those elements that are in a but not in b. The set b in 17 is formed from the elements that are already in u. The set of all x in u such that x is an even number the set u is called the universe. Usually we denote sets with uppercase letters, elements with lowercase letters.
Set theory proof with cartesian product of sets and intersection a x b n c a x b n a x c broll. Set theory sets a set is a collection of objects, called its elements. Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. Set operations and venn diagrams university of hawaii. Introduction to modern set theory a to z directory. B read a union b or the union of a and b is defined as the set that consists of all elements belonging to either set a or set b or both. Set theory basic set theory stanford encyclopedia of. Goldrei classic set theory, chapman and hall 1996, or h. 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 always false then the implication is always true.
Chapter 4 set theory \ a set is a many that allows itself to be thought of as a one. Nb note bene it is almost never necessary in a mathematical proof to remember that a function is literally a set of ordered pairs. Then by the axiom schema of comprehension, there is a set bdf x2vw g. Introduction to logic and set theory 202014 general course notes december 2, 20 these notes were prepared as an aid to the student. A since every element of b is also an even number, so is an element of a. If x is a subset of the real numbers, then either there is a onetoone function from the set of real numbers into x or there is a onetoone. Some sets occur so frequently that there are standard names and symbols for them. If aand bare sets, we say that bis a subset of aif every element of b is an element of a. Then the complement of a is the set of all elements of u which are not the elements of a. What can we say about set a if x a for all objects x. With setbuilder notation, we can list a few more very important sets. Basic set theory lx 502 semantics i september 11, 2008 1. Set theory if we are interested in elements of a set a that are not contained in a set b, we can write this set as a.
I wrote it in the rm belief that set theory is good not just for set theorists, but for many mathematicians, and that the earlier a student sees the particular point of view that we call modern set theory, the better. Review of set theory michael williams last updated. Sets and operations on sets mathematics libretexts. For any two sets x and y, either there is a onetoone function from. That is, if an element belongs to set aor set bthen it belongs to the union of a and b. Here are three simple statements about sets and functions. A binary operation is called commutative if the order of the things it operates on doesnt matter. B, is the set of all elements that belong to a,ortob, or to both.
The basic relation in set theory is that of elementhood, or membership. For those of you new to abstract mathematics elementary does not mean simple though much of the material. In mathematics, specifically set theory, the cartesian product of two sets a and b, denoted a. Creating new sets set operations binary operations the union of two sets a and b is denoted a. Set theory proof with cartesian product of sets and intersection a x. To indicate that an object x is a member of a set a one writes x. A b x x a x b the complement of a is the complement of a with respect to u. You must have also heard of subset and superset, which are the.
March 30, 2009 basic set theory we will use the standard notation for containments. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. But in the early 1900s, a radical transformation occurred in mathematiciansunderstanding of sets. C is not a subset of a, since c contains an element, 3, that is not contained in a.
Set theory is indivisible from logic where computer science has its roots. Know about the important set theory rules and check the set theory examples based on concepts set theory. Set theory is the mathematical theory of welldetermined collections, called sets, of objects that are called members, or elements, of the set. The difference of a and b is also called the complement of b with respect to a. Set symbols of set theory and probability with name and definition. If a is an element of a set a, we say that a belongs to a the greek symbol. Set theory for computer science university of cambridge. 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.
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