Set Notation
Set notation is a mathematical language used to describe and manipulate collections of objects, known as sets. In set notation, sets are often denoted by capital letters and their elements are listed within curly brackets, such as L = {television, sofa, coffee table, bookcase, lamp}. This notation can represent both finite sets, which contain a limited number of elements, and infinite sets, defined by specific criteria, such as {y | y > 4}, which includes all values of y greater than 4. Set notation is fundamental in various areas of mathematics, including algebra, calculus, and statistics, enabling mathematicians to discuss relationships and operations among different sets. For instance, one can express that a set A is a subset of a set B, indicating that all elements in A are also included in B.
Set theory, significantly developed by Georg Cantor, addresses the properties and relationships of sets, including various types of numbers like integers, rational numbers, and real numbers. Cantor's work on infinite sets, particularly his exploration of different sizes of infinity, revolutionized mathematical understanding by raising questions about the nature of infinite collections. Overall, set notation serves as a crucial tool for mathematicians to articulate and explore complex mathematical concepts.
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Set Notation
In broadest terms, a set is a collection of things, with the individual things within the set called either elements or members of the set. Mathematicians usually speak of sets as a collection of numbers, but a set may be defined with ordinary objects. Sets are usually referenced by an italicized capital letter, with the object in the set listed between a pair of curly brackets. A set of objects in a living room, L, might be defined as L = {television, sofa, coffee table, bookcase, lamp}. In this case, the set L is a finite set that contains five elements. This set notation is used throughout mathematics to describe the elements of sets and how sets relate to each other.
Rather than listing out individual elements within a set, some sets contain all elements that fit a given rule or criterion. An equation with a solution of y > 4 can be written using set notation as {y|y > 4}, which is read as "the set of all y, such that y is greater than 4." This is an infinite set, because it contains an infinite number of elements.
Overview
Set notation is ubiquitous in mathematics, because it provides mathematicians with a way to discuss relationships among groups of numbers. Set notation can be used in algebra and calculus to describe sets of numbers. Functions are defined in terms of a mapping from one set of inputs (the domain) to a set of outputs (the range). In statistics and probability, set notation is often used to designate the sample space of interest.
Having defined a pair of sets A and B, the mathematician is then able to use set notation to designate different ways of combining and relating sets to each other. For example, A could be a subset of B, which indicates that every element of A is also an element of B. Set notation for indicating that A is a subset of B is
.
The use of sets developed into the field of set theory under Georg Cantor (1845–1918). By defining sets that have common features, it is possible for mathematicians to deeply explore fundamental relationships within mathematics. The importance of this approach to understanding deep truths about mathematics becomes more obvious when you consider some relevant sets of numbers.
: the set of all counting numbers
: the set of all integers
: the set of all rational numbers
: the set of all real numbers
These sets all contain an infinite number of elements, but the question that Cantor posed was built around the idea of whether all infinite sets contained the same number of infinities. For example, there are an infinite number of integers, but between any two integers are also an infinite number of rational numbers. Cantor wanted to know if this meant that the collection of rational numbers was a larger infinity than the collection of integers, and he applied set theory to this problem. Ultimately, his exploration of these sets led to a revolution in how mathematicians viewed the concept of infinity.
Bibliography
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Berlinski, David. "Sets." Infinite Ascent: A Short History of Mathematics. New York, NY: Modern Library, 2005. Print.
Devlin, Keith. J. The Joy of Sets: Fundamentals of Contemporary Set Theory. 2nd ed. New York: Springer, 1993. Print.
Elwes, Richard. "Cardinal Numbers." Math in 100 Key Breakthroughs. New York: Quercus, 2013. Print.
Strogatz, Steven. "The Hilbert Hotel." The Joy of X. Boston: Houghton, 2012. Print.
Wallace, David Foster. Everything and More: A Compact History of Infinity. New York: Norton, 2010.