Chap 1 The Number System

# The rst collection of properties of r is generally

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: umber. The ﬁrst collection of properties of R is generally known as the Field axioms. We oﬀer no proof of these properties, and simply treat and accept them as given. FIELD AXIOMS. (A1) For every a, b ∈ R, we have a + b ∈ R. (A2) For every a, b, c ∈ R, we have a + (b + c) = (a + b) + c. (A3) For every a ∈ R, we have a + 0 = a. (A4) For every a ∈ R, there exists −a ∈ R such that a + (−a) = 0. (A5) For every a, b ∈ R, we have a + b = b + a. (M1) For every a, b ∈ R, we have ab ∈ R. (M2) For every a, b, c ∈ R, we have a(bc) = (ab)c. (M3) For every a ∈ R, we have a1 = a. (M4) For every a ∈ R such that a = 0, there exists a−1 ∈ R such that aa−1 = 1. (M5) For every a, b ∈ R, we have ab = ba. (D) For every a, b, c ∈ R, we have a(b + c) = ab + ac. Remark. The properties (A1)–(A5) concern the operation addition, while the properties (M1)–(M5) concern the operation multiplication. In the terminology of group theory, not usually covered in ﬁrst Chapter 1 : The Number System page 1 of 19 First Year Calculus c W W L Chen, 1982, 2005 year mathematics, we say that the set R forms an abelian group under addition, and that the set...
View Full Document

## This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.

Ask a homework question - tutors are online