chapter1 - Chapter 1 The Field of Reals and Beyond Our goal...

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Unformatted text preview: Chapter 1 The Field of Reals and Beyond Our goal with this section is to develop (review) the basic structure that character- izes the set of real numbers. Much of the material in the rst section is a review of properties that were studied in MAT108 however, there are a few slight differ- ences in the de nitions for some of the terms. Rather than prove that we can get from the presentation given by the author of our MAT127A textbook to the previous set of properties, with one exception, we will base our discussion and derivations on the new set. As a general rule the de nitions offered in this set of Compan- ion Notes will be stated in symbolic form this is done to reinforce the language of mathematics and to give the statements in a form that clari es how one might prove satisfaction or lack of satisfaction of the properties. YOUR GLOSSARIES ALWAYS SHOULD CONTAIN THE (IN SYMBOLIC FORM) DEFINITION AS GIVEN IN OUR NOTES because that is the form that will be required for suc- cessful completion of literacy quizzes and exams where such statements may be requested. 1.1 Fields Recall the following DEFINITIONS: The Cartesian product of two sets A and B , denoted by A B , is a b : a + A F b + B . 1 2 CHAPTER 1. THE FIELD OF REALS AND BEYOND A function h from A into B is a subset of A B such that (i) 1 a [ a + A " 2 b b + B F a b + h ] i.e., dom h A , and (ii) 1 a 1 b 1 c [ a b + h F a c + h " b c ] i.e., h is single-valued. A binary operation on a set A is a function from A A into A . A eld is an algebraic structure, denoted by I e f , that includes a set of objects, I , and two binary operations, addition and multiplication , that satisfy the Axioms of Addition, Axioms of Multiplication, and the Distributive Law as described in the following list. (A) Axioms of Addition ( I e is a commutative group under the binary operation of addition with the additive identity denoted by e ) (A1) : I I I (A2) 1 x 1 y x y + I " x y y x (commutative with respect to addition) (A3) 1 x 1 y 1 z b x y z + I " d x y z x y z ec (asso- ciative with respect to addition) (A4) 2 e [ e + I F 1 x x + I " x e e x x ] (additive identity property) (A5) 1 x x + I " 2 x [ x + I F x x x x e ] (additive inverse property) (M) Axioms of Multiplication ( I f is a commutative group under the binary operation of multiplication with the multiplicative identity denoted by f ) (M1) : I I I (M2) 1 x 1 y x y + I " x y y x (commutative with respect to multiplication) (M3) 1 x 1 y 1 z b x y z + I " d x y z x y z ec (associative with respect to multiplication) (M4) 2 f d f + I F f / e F 1 x x + I " x f f x x e (mul- tiplicative identity property) (M5) 1 x x + I e " db 2 b x 1 ccb x 1 + I F x x 1 x 1 x f ce (multiplicative inverse property) 1.1. FIELDS 3 (D) The Distributive Law 1 x 1 y 1 z x y z + I " [ x y z x y x z ] Remark 1.1.1 Properties (A1) and (M1) tell us that I is closed under addition and closed under multiplication, respectively....
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chapter1 - Chapter 1 The Field of Reals and Beyond Our goal...

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