C. Determine if the specified operation is a binary operation on the given set. Justify your answer. 1) The operation subtraction on the set of natural numbers N. 2) The operation defined by a b = on the set of rational numbers Q except 0. 3) The operation * defined by a b = +vab on the set of real numbers R. %3D

C. Determine if the specified operation is a binary operation on the given set. Justify your answer. 1) The operation subtraction on the set of natural numbers N. 2) The operation defined by a b = on the set of rational numbers Q except 0. 3) The operation * defined by a b = +vab on the set of real numbers R. %3D

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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Concept explainers

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

100%

C. Determine if the specified operation is a binary operation on the given set. Justify
your answer.
1) The operation subtraction on the set of natural numbers N.
2) The operation * defined by a
on the set of rational numbers Q except 0.
%3D
3) The operation defined by a b = ±vab on the set of real numbers R.
%3D

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Another Field We'll let Q(√2) represent the set of numbers that can be written in the form a+b√2, where 'a' and 'b' and rational numbers (i.e., 'a' and 'b' are from 13 2 the set Q). For example, +¹3√2, 7.2-6√2, and 15 (= 15 + 0√2) are three numbers that would be in the set Q(√2), but √√3+ 4√2 or 3 + √2 would not be in the set Q(√2) (neither √√3 nor are rational numbers). A number such 3+5√2 would be in the set Q(√2) since 3+√2 can be re-written in the form 7 a+b√2 using the following steps: 7 7 3-5√2 21-35√2 21-35√2 21 - 35√2 21 35 == + √2 3+5√2 3+5√2 3-5√2 9 - 25(2) -41 41 41 = * Your task: Show that the system (Q(√2), +, *) is a field, where + is the ordinary addition operation and * is the ordinary multiplication operation. So, you need to show that: a. (Q(√2), +, *) is a commutative ring, and b. (Q(√2) - {0}, *) is a group.