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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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part 4 5 6
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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