4055bckgrnd

4055bckgrnd - Math 4055 Background Notes Ambar N. Sengupta...

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Unformatted text preview: Math 4055 Background Notes Ambar N. Sengupta Department of Mathematics Louisiana State University, Baton Rouge February, 2002 2 Chapter 1 Set Theory Set theory is both a branch of mathematics and, along with logic, the foun- dation for mathematics. It is the structure on which all of mathematics rests and provides the language in which practically all of mathematics is written. Virtually every object in mathematics is a set. Set theory has no applications in daily life, nor does it have any direct application to the sciences. However, it is the foundation for all of mathe- matics and thus, indirectly, provides the solid ground on which an enormous amount of science, engineering, statistics and finance rests. No one invented set theory with the intention of providing the foundation for mathematics, let alone for any concrete applications in science or daily life. Set theory was invented by Georg Cantor in the 1860s as a systematic framework in which he could formulate his abstruse investigations of infinite numbers. His ideas were considered too esoteric then and few of the great minds of that era recognized that Cantor had invented an amazingly versatile language which could be used to formulate virtually every mathematical concept in a precise way. Cantors theory eventually led to the development of mathematical logic and then this led much later, in the twentieth century, through the works of G odel and Turing, to the idea of a machine which could execute commands without needing to understand (in human intuitive terms) what the commands meant. Eventually, such a machine was indeed built and today we know it as the computer. Most working mathematicians today view set theory as an enormously useful language in which they write their works. But many consider investi- gation of set theory in itself an abstruse endeavor. However, set theory is in fact still very much an active area of research with many deep questions. 3 4 CHAPTER 1. SET THEORY 1.1 Sets A set is a collection of objects. These objects are called elements of the set. We often write a set by displaying its elements within braces. For exam- ple, { a, b, c } is the set whose elements are a , b , and c . It is important to note that the set { a, b, c } is the same as the set { b, c, a } ; these being just two different ways of displaying the same set. Similarly, { a, b, b } is the same set as { a, b } , since both sets really have the exact same elements a and b . A set x is said to be equal to a set y , written x = y , if x and y have exactly the same elements. For example, { a, a, b, b, c } = { b, a, c } because both sets have exactly the same elements: a , b , and c . To prove that two sets are not equal we need only produce an object which is an element of one set but not an element of the other ....
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4055bckgrnd - Math 4055 Background Notes Ambar N. Sengupta...

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