Exercise 1. Prove that m 1 = m. (Hint: it is not necessary to argue by induction on m.) Exercise 2. Prove the following statements: (a) If A is a set, then A is transitive iff A C P(A). (b) If A is a transitive set, then P(A) is a transitive set. (c) If B is a set of transitive sets, then UB is a transitive set. Exercise 3. Prove that for all m Ew, either m = Ø or ØЄ m. (Hint: Show that S = {m Є w | m = 0 or 0 Є m} is inductive.)

Exercise 1. Prove that m 1 = m. (Hint: it is not necessary to argue by induction on m.) Exercise 2. Prove the following statements: (a) If A is a set, then A is transitive iff A C P(A). (b) If A is a transitive set, then P(A) is a transitive set. (c) If B is a set of transitive sets, then UB is a transitive set. Exercise 3. Prove that for all m Ew, either m = Ø or ØЄ m. (Hint: Show that S = {m Є w | m = 0 or 0 Є m} is inductive.)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 13E: Assume that is an associative binary operation on the non empty set A. Prove that a[ b(cd) ]=[...

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