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Introduction – Sets of Numbers (9/4)

Introduction – Sets of Numbers (9/4). Z - integers Z + - positive integers Q - rational numbers Q + - positive rationals R - real numbers R + - positive reals C - complex numbers

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Introduction – Sets of Numbers (9/4)

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  1. Introduction – Sets of Numbers (9/4) • Z - integers Z+ - positive integers • Q - rational numbers Q+ - positive rationals • R - real numbers R+ - positive reals • C - complex numbers • For any number set S, by S* we mean the set with 0 removed. So, for example, Q* means all non-zero rationals. • Zn - the set of numbers {0, 1, 2, ..., n – 1} • U(n) - the subset of Zn consisting of numbers which are relatively primeto n. • For example, what is U(12)? What is U(13)?

  2. Algebraic Objects • Any set which has one or more binary operations on it is called an algebraic object. (A binary operation on a set combines two elements of the set to produce a third element of the set. For example, R has 4 binary operations but Z has only 3. What are they and why?) • Abstract Algebra is the study of algebraic objects, both from a general, abstract point of view and from looking at many examples. • There are many types of abstract algebraic objects: groups, rings, fields, vector spaces, modules, etc. • In this course, we concentrate on groupssince in some ways they are the simplest.

  3. Loose Definition of a Group • We will be somewhat more precise shortly, but for the moment we consider the following definition: • A group(G, )is a set G possessing a single binary operation  such that: • (Existence of an identity element) There exists an element e in G such that for every a  G, a  e = e  a = a. • (Existence of inverses) For every element a G, there exists an element a-1  G such that a  a-1= a-1 a = e. • When working with an abstract group G, we often omit the symbol  and simply use “juxtaposition” (i.e., write a b in place of a b).

  4. Simple Example of a Group • Consider the set Z and operation +. • Is + a binary operation on Z? • Does there exist an identity element for + in Z? If so, what is it? • Given in element a  Z (i.e., given any integer), is there another element a-1  Z such that a + a-1 = the identity element? If so, what is it? • So, is (Z, +) a group?

  5. Is it a group? Yes (A) or No (B) • (Z+, +) (i.e., positive integers under addition) • (Z, .) (i.e., the integers under multiplication) • (2Z, +) (i.e., the even integers under addition) • (Q, +) • (Q, .) • (Q*, .) • (R[x], +) (i.e., all polynomials with real coefficients under +) • All 2 by 2 matrices with coefficients in Q under matrix multiplication. • (Z12, +12) (i.e., Z12 under “addition mod 12”) • (Z12*, .12)(i.e., Z12 under “multiplication mod 12”)

  6. Assignment for Friday • Obtain the text. • Do the follow-up assignment (not to hand in).

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