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Determine if the following sets G with the operation indicated form a group. If not, point out which of the group axioms fail. (a) G set of all integers, a * b = a - b. = (b) G = set of all integers, a * b = a + b + ab. (c) G = set of nonnegative integers, a * b = a + b.v (d) G = set of all rational numbers # -1, a*b = a + b + ab./ (e) G = set of all rational numbers with denominator divisible by 5 (writ- ten so that numerator and denominator are relatively prime), a * b = a + b. J (f) G a set having more than one element, a * b = a for all a, b E G.

Determine if the following sets G with the operation indicated form a group. If not, point out which of the group axioms fail. (a) G set of all integers, a * b = a - b. = (b) G = set of all integers, a * b = a + b + ab. (c) G = set of nonnegative integers, a * b = a + b.v (d) G = set of all rational numbers # -1, a*b = a + b + ab./ (e) G = set of all rational numbers with denominator divisible by 5 (writ- ten so that numerator and denominator are relatively prime), a * b = a + b. J (f) G a set having more than one element, a * b = a for all a, b E G.

Advanced Engineering Mathematics
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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Determine if the following sets G with the operation indicated form a group. If not, point out which of the group axioms fail. (a) ´G = set of all integers, a * b = a = b. (b) G set of all integers, a * b = a + b + ab. = (c) G = set of nonnegative integers, a * b = a + b.v (d) G = set of all rational numbers + -1, a*b = a + b + ab./ (e) G= set of all rational numbers with denominator divisible by 5 (writ- ten so that numerator and denominator are relatively prime), a * b = a + b. (f) G a set having more than one element, a * b = a for all a, b E G. L
Transcribed Image Text:Determine if the following sets G with the operation indicated form a group. If not, point out which of the group axioms fail. (a) ´G = set of all integers, a * b = a = b. (b) G set of all integers, a * b = a + b + ab. = (c) G = set of nonnegative integers, a * b = a + b.v (d) G = set of all rational numbers + -1, a*b = a + b + ab./ (e) G= set of all rational numbers with denominator divisible by 5 (writ- ten so that numerator and denominator are relatively prime), a * b = a + b. (f) G a set having more than one element, a * b = a for all a, b E G. L
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