Set theory cardinality pdf

Cantorian set theory is founded on the principles of extension and abstraction, described above. Sets and elements set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. These notes were prepared using notes from the course taught by uri avraham, assaf hasson, and of course, matti rubin. Set symbols of set theory and probability with name and definition. The two sets a 1,2,3 and b a,b,c thus have the cardinality since we can match up the elements of the two sets in such a way that each element in each set is matched with exactly one element in the other set. The axioms of set theory, ordinal and cardinal arithmetic, the axiom of foundation, relativisation, absoluteness, and reflection, ordinal definable sets and inner models of set theory, the constructible universe l cohens method of forcing, independence. A set is a collection of distinct objects, called elements of the set. This is easy to prove via cantors diagonalization and it sidesteps the reification problem for cardinalities, i.

It says that the relation \can be put into 11 correspondence is transitive. In mathematics, the cardinality of a set means the number of its elements. Then by the axiom schema of comprehension, there is a set bdf x2vw g. So we have a disjoint union of two nonempty sets is empty, or that cardinality zero is no longer the unique property of the empty set. Introductory cardinality theory alan kaylor cline although by name the theory of set cardinality may seem to be an offshoot of combinatorics, the central interest is actually infinite sets. Two sets are equal if and only if they have the same elements. There are two ways to define the cardinality of a set. They have the same number of elements if and only if there is a 1to1 correspondence between the sets. This is usually taken as the definition of cardinal number in axiomatic set theory.

To indicate that an object x is a member of a set a one writes x. This alone assures the subject of a place prominent in human culture. S contradiction but every element must either be in s or not in s. Unless you plan on studying advanced set theory or the foundations. A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets.

I believe this is appropriate for a general mathematical audience. Set theory is one of the most fundamental branch of mathematics, but is also also very complex if you try to analyze three or more sets. These elements could be numbers, alphabets, variables, etc. It will take the development of some theory before this statement can be made meaningful.

The idea is that when comparing the sizes of two sets a and b, we attempt to. It has been and is likely to continue to be a a source of fundamental ideas in computer science from theory to practice. Set theory \a set is a many that allows itself to be thought of as a one. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces. 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. Cardinality the cardinality of a set is roughly the number of elements in a set. The set of all books written about travel to chile. Some examples of sets defined by describing the contents. For those of you new to abstract mathematics elementary does not mean simple though much of the material. The resulting axiomatic set theory became known as zermelofraenkel zf set theory. In informal set theory one often represents sets as possibly overlapping disks in the plane and interprets these operations geometrically this is. In a typical undergraduate level textbook on set theory, a set is defined as a collection of things called its members or elements, the collection being regarded as a single object enderton, 1977, p. The order of the elements in a set doesnt contribute.

They are not guaranteed to be comprehensive of the material covered in the course. The number of elements in a set is called the cardinality of the set. Cardinality is the settheoretic notion of the size or number of elements of a set. I can tell that two sets have the same number of elements by trying to pair the elements up. As will be seen, the tool employed for the majority of the theory is establishing one. This means that 1,2,3 is a set but 1,1,3 is not because 1 appears twice in the second collection. Cardinality after counting, when asked how many there are in the set, a student. The order of the elements in a set doesnt contribute anything new. One might say that set theory was born in late 1873, when he made the amazing discovery that the linear continuum, that is, the real line, is not countable, meaning that its points cannot be counted using the natural numbers. The cardinality relation thus partitions the set of all sets to. If a has only a finite number of elements, its cardinality is simply the number of elements in a.

T is the set of all juices in the market without added sugar. A set is a collection of objects satisfying a certain set. Also known as the cardinality, the number of distinct elements within a set provides a. For example, if s is the set of all juices in the supermarket, and t is the set of all foodstu. Computer science, being a science of the arti cial, has had many of its constructs and ideas inspired by set theory. This means that cardinality, when extended to negative integers, cannot extend the usual arithmetics as well. This note is an introduction to the zermelofraenkel set theory with choice zfc. Show that countable infinity is the smallest infinite cardinality, i. Ling 310, adapted from umass ling 409, partee lecture notes march 1, 2006 p. Why is the cardinality of real numbers equal to the power set. To describe some results based upon these principles, the notion of equivalence of sets will be defined.

Georg cantor this chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical functions. The notation and symbols for sets are based on the operations performed on them. But even more, set theory is the milieu in which mathematics takes place today. In naive set theory, a set is a collection of objects called members or elements that is regarded as being a single object. Cardinality simple english wikipedia, the free encyclopedia. This third article further compounds this knowledge by zoning in on the most important property of any given set. A set may be defined by a membership rule formula or by listing its. Set theory, as a separate mathematical discipline, begins in the work of georg cantor. For example, the set a 2, 4, 6 contains 3 elements, and therefore a has a cardinality of 3. Cantor initiated the study of set theory with his investigations on the cardinality of sets of real numbers.

Gives the wrong number recounts to determine the number of items comparing number number conservation when asked to compare two sets and identify which has more, for example, a student. Set theory for computer science university of cambridge. Introduction to logic and set theory202014 general course notes december 2, 20 these notes were prepared as an aid to the student. Aug 05, 2015 the cardinality of math\sigmamath can never be the same as that of math\mathcalp\sigmamath, since a fundamental theorem about cardinalities of sets is that the cardinality of math\mathcalpsmath is always strictly greater. In maths, set theory was developed to explain about collections of objects. The axiom of pair, the axiom of union, and the axiom of. Two sets have the same or equal cardinality if they have the same number of elements. Sets a and b have the same cardinality if there is a bijection between them for fnite sets, cardinality is the number of elements there is a bijection between nelement set a and 1, 2, 3, n following ernie croots slides. Set theory is indivisible from logic where computer science has its roots. Basic set theory a set is a many that allows itself to be thought of as a one. A representative set is designated for each equivalence class. Before discussing infinite sets, which is the main discussion of this section, we would like to talk about a very useful rule. In particular, he proved that there are dif ferent infinite. For finite sets, cardinalities are natural numbers.

The idea is that two sets are equivalent if it is possible to pair off members of the first set with members of the second, with no leftover members on. Set theory begins with a fundamental binary relation between an object o and a set a. Set theory is a branch of mathematics which deals with the study of sets or the collection of similar objects. Although by name the theory of set cardinality may seem to be an offshoot of combinatorics, the central interest is actually infinite sets. For example, we can match 1 to a, 2 to b, or 3 to c. Free set theory books download ebooks online textbooks. Duplicates dont contribute anythi ng new to a set, so remove them. Sets and cardinality notes for 620111 school of mathematics and. Basic concepts of set theory, functions and relations. The cardinality of a set a is defined as its equivalence class under equinumerosity. In zermelofraenkel without requiring the axiom of choice zf, a common model for set theory, this means that aleph1 is the successor cardinality to aleph0, in the sense that aleph0 set theory formulas. 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.