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We define R to be the set in which two operations are defined, + : R x R + R, and x : Rx R + R. and such that (R, +, x) satisfies the following axioms: Field arioms: (i) the commutative property of the addition and multiplication operations: a+b = b+ a; ab = ba; (ü) the associative property of the addition and multiplication operations: (a + b) +c = a+ (b+c); (ab)e = a (be); (ii) the distributive property of products over sums (a + b) c = ac + be, a (b+ c) = ab+ ac; (iv) the existence of an additive identity 0 and a distinct multiplicative iden- tity, that is, 1 + 0, satisfying a +0 = a for all a E R and a x1 = a for all a € R; (v) for each element a E R, the existence of an additive inverse b, also called opposite, i.e an element b such that a+ b = 0; (vi) for each element a + 0 there exists a multiplicative inverse b, ie. an element b such that a x b = 1; Ordering arioms: There exists a subset of R, which we denote R* which does not contain 0, which satisfies the following properties: (vii) R* is closed under the operations of sum and multiplication, that is: a, be Rt = a+ b, ab € Rt; (viii) for any a ER, either a E R+, or -a e R*, or a = 0 (ir) for a, b, c € R, a – bER* and b – e eR* = a - ceRt. Besides the field axioms ((i)-(vi) and the ordering axioms (vii)-(iz), which are also satisfied by the set of rational numbers Q, the real numbers also satisfy (x) the completeness ariom.

We define R to be the set in which two operations are defined, + : R x R + R, and x : Rx R + R. and such that (R, +, x) satisfies the following axioms: Field arioms: (i) the commutative property of the addition and multiplication operations: a+b = b+ a; ab = ba; (ü) the associative property of the addition and multiplication operations: (a + b) +c = a+ (b+c); (ab)e = a (be); (ii) the distributive property of products over sums (a + b) c = ac + be, a (b+ c) = ab+ ac; (iv) the existence of an additive identity 0 and a distinct multiplicative iden- tity, that is, 1 + 0, satisfying a +0 = a for all a E R and a x1 = a for all a € R; (v) for each element a E R, the existence of an additive inverse b, also called opposite, i.e an element b such that a+ b = 0; (vi) for each element a + 0 there exists a multiplicative inverse b, ie. an element b such that a x b = 1; Ordering arioms: There exists a subset of R, which we denote R* which does not contain 0, which satisfies the following properties: (vii) R* is closed under the operations of sum and multiplication, that is: a, be Rt = a+ b, ab € Rt; (viii) for any a ER, either a E R+, or -a e R*, or a = 0 (ir) for a, b, c € R, a – bER* and b – e eR* = a - ceRt. Besides the field axioms ((i)-(vi) and the ordering axioms (vii)-(iz), which are also satisfied by the set of rational numbers Q, the real numbers also satisfy (x) the completeness ariom.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.1: Definition Of A Group
Problem 34E: 34. Let be the set of eight elements with identity element and noncommutative multiplication...
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We define R to be the set in which two operations are defined, + : Rx R + R, and x : Rx R + R. and such that (R, +, x) satisfies the following axioms: Field arioms: (i) the commutative property of the addition and multiplication operations: a +b = b+ a; ab = ba; (ü) the associative property of the addition and multiplication operations: (a + b) +c = a+ (b+c); (ab) c = a (be); (ii) the distributive property of products over sums (a + 6) c = ac + be, a (b+ c) = ab+ ac; (iv) the existence of an additive identity 0 and a distinct multiplicative iden- tity, that is, 1 + 0, satisfying a +0 = a for all a E R and a x1 = a for all a € R; (v) for each element a € R, the existence of an additive inverse b, also called opposite, i.e an element b such that a + b = 0; (vi) for each element a + 0 there exists a multiplicative inverse b, ie. an element b such that a x b = 1; Ordering arioms: There exists a subset of R, which we denote R* which does not contain 0, which satisfies the following properties: (vii) R* is closed under the operations of sum and multiplication, that is: a, be R+ = a+ b, ab € Rt; (viii) for any a ER, either a eR+, or -a e R*, or a = 0 (ir) for a, b, e e R, a -be Rt and 6-CER* = a -cERt. Besides the field axioms ((i)-(vi) and the ordering axioms (vii)-(iz), which are also satisfied by the set of rational mumbers Q, the real numbers also satisfy (x) the completeness ariom.
Transcribed Image Text:We define R to be the set in which two operations are defined, + : Rx R + R, and x : Rx R + R. and such that (R, +, x) satisfies the following axioms: Field arioms: (i) the commutative property of the addition and multiplication operations: a +b = b+ a; ab = ba; (ü) the associative property of the addition and multiplication operations: (a + b) +c = a+ (b+c); (ab) c = a (be); (ii) the distributive property of products over sums (a + 6) c = ac + be, a (b+ c) = ab+ ac; (iv) the existence of an additive identity 0 and a distinct multiplicative iden- tity, that is, 1 + 0, satisfying a +0 = a for all a E R and a x1 = a for all a € R; (v) for each element a € R, the existence of an additive inverse b, also called opposite, i.e an element b such that a + b = 0; (vi) for each element a + 0 there exists a multiplicative inverse b, ie. an element b such that a x b = 1; Ordering arioms: There exists a subset of R, which we denote R* which does not contain 0, which satisfies the following properties: (vii) R* is closed under the operations of sum and multiplication, that is: a, be R+ = a+ b, ab € Rt; (viii) for any a ER, either a eR+, or -a e R*, or a = 0 (ir) for a, b, e e R, a -be Rt and 6-CER* = a -cERt. Besides the field axioms ((i)-(vi) and the ordering axioms (vii)-(iz), which are also satisfied by the set of rational mumbers Q, the real numbers also satisfy (x) the completeness ariom.
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