Say that X ⊆Ris invariant under successors if for all x ∈R, we have that x ∈X if and only if x + 1 ∈X. a. Which of the following sets are invariant under successors? [Explain your answer, that is, prove or disprove that the given set is invariant under successors.] • The negative integers; • The positive integers; • The set of all irrational numbers. b. Prove or disprove the following statement: • Suppose X, Y ⊆Rare invariant under successors. If X ⊆Y then X = Y .

Say that X ⊆Ris invariant under successors if for all x ∈R, we have that x ∈X if and only if x + 1 ∈X. a. Which of the following sets are invariant under successors? [Explain your answer, that is, prove or disprove that the given set is invariant under successors.] • The negative integers; • The positive integers; • The set of all irrational numbers. b. Prove or disprove the following statement: • Suppose X, Y ⊆Rare invariant under successors. If X ⊆Y then X = Y .

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 9E: 9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of...

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