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(1) Let G = R \ {0}. the set of non-zero real numbers. Prove that G, together with the operation * given by xy=2xy for all x, y Є G, is a group. Your answer should be written as a formal proof (see Question 3.2 in the Course Notes for a similar example). As usual, you may assume that multiplication of real numbers is associative (brackets do not change the answer) and commutative (reordering products does not change the answer). Hint: To prove this, you need to prove that the four axioms hold, which, for this example, are: (G1) Closure: For all x, y = R \ {0}, x * y = R \ {0}. (G2) Associativity: For all x, y, z Є R \ {0}, (x * y) * z = x * (y * z). (G3) Identity: There is an element e Є R \ {0} such that, for all x R \ {0}, x * e = = x and e* x = x. (G4) Inverses: For all x = R \ {0}, there is an element y EЄ R \ {0} such that x * y = e and y* x = e. 9 marks: 2 for each axiom, 1 for conclusion

(1) Let G = R \ {0}. the set of non-zero real numbers. Prove that G, together with the operation * given by xy=2xy for all x, y Є G, is a group. Your answer should be written as a formal proof (see Question 3.2 in the Course Notes for a similar example). As usual, you may assume that multiplication of real numbers is associative (brackets do not change the answer) and commutative (reordering products does not change the answer). Hint: To prove this, you need to prove that the four axioms hold, which, for this example, are: (G1) Closure: For all x, y = R \ {0}, x * y = R \ {0}. (G2) Associativity: For all x, y, z Є R \ {0}, (x * y) * z = x * (y * z). (G3) Identity: There is an element e Є R \ {0} such that, for all x R \ {0}, x * e = = x and e* x = x. (G4) Inverses: For all x = R \ {0}, there is an element y EЄ R \ {0} such that x * y = e and y* x = e. 9 marks: 2 for each axiom, 1 for conclusion

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 9E: Exercises In Exercises, decide whether each of the given sets is a group with respect to the...
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(1) Let G = R \ {0}. the set of non-zero real numbers. Prove that G, together with the operation * given by xy=2xy for all x, y Є G, is a group. Your answer should be written as a formal proof (see Question 3.2 in the Course Notes for a similar example). As usual, you may assume that multiplication of real numbers is associative (brackets do not change the answer) and commutative (reordering products does not change the answer). Hint: To prove this, you need to prove that the four axioms hold, which, for this example, are: (G1) Closure: For all x, y = R \ {0}, x * y = R \ {0}. (G2) Associativity: For all x, y, z Є R \ {0}, (x * y) * z = x * (y * z). (G3) Identity: There is an element e Є R \ {0} such that, for all x R \ {0}, x * e = = x and e* x = x. (G4) Inverses: For all x = R \ {0}, there is an element y EЄ R \ {0} such that x * y = e and y* x = e. 9 marks: 2 for each axiom, 1 for conclusion
Transcribed Image Text:(1) Let G = R \ {0}. the set of non-zero real numbers. Prove that G, together with the operation * given by xy=2xy for all x, y Є G, is a group. Your answer should be written as a formal proof (see Question 3.2 in the Course Notes for a similar example). As usual, you may assume that multiplication of real numbers is associative (brackets do not change the answer) and commutative (reordering products does not change the answer). Hint: To prove this, you need to prove that the four axioms hold, which, for this example, are: (G1) Closure: For all x, y = R \ {0}, x * y = R \ {0}. (G2) Associativity: For all x, y, z Є R \ {0}, (x * y) * z = x * (y * z). (G3) Identity: There is an element e Є R \ {0} such that, for all x R \ {0}, x * e = = x and e* x = x. (G4) Inverses: For all x = R \ {0}, there is an element y EЄ R \ {0} such that x * y = e and y* x = e. 9 marks: 2 for each axiom, 1 for conclusion
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