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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...
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