Let A be a non-empty set, ≃ ⊆ A × A an equivalence realtionship and ⪯ ⊆ A × A a partial order, both over A. Consider the quotient set: A / ≃, and define the following relation ≪ ⊆ (A/ ≃) × (A/ ≃) where (S1, S2) ∈ ≪ if, and only if, there is an a ∈ S1 such that for all b ∈ S2, a ⪯ b. 1.) How to show that ≪r is is a partial order over A/ ≃ where ≪r is the reflex cause of ≪. 2.) Is it true that A has a minimal element according to ⪯ if, and only if, A/ ≃ has a minimal element according to ≪r ? How to show if it is true or false (in such case suggest a counterexample)
Let A be a non-empty set, ≃ ⊆ A × A an equivalence realtionship and ⪯ ⊆ A × A a partial order, both over A. Consider the quotient set: A / ≃, and define the following relation ≪ ⊆ (A/ ≃) × (A/ ≃) where (S1, S2) ∈ ≪ if, and only if, there is an a ∈ S1 such that for all b ∈ S2, a ⪯ b. 1.) How to show that ≪r is is a partial order over A/ ≃ where ≪r is the reflex cause of ≪. 2.) Is it true that A has a minimal element according to ⪯ if, and only if, A/ ≃ has a minimal element according to ≪r ? How to show if it is true or false (in such case suggest a counterexample)
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 21E: 21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in...
Related questions
Question
Let A be a non-empty set, ≃ ⊆ A × A an equivalence realtionship and ⪯ ⊆ A × A a partial order, both over A. Consider the quotient set: A / ≃, and define the following relation ≪ ⊆ (A/ ≃) × (A/ ≃) where (S1, S2) ∈ ≪ if, and only if, there is an a ∈ S1 such that for all b ∈ S2, a ⪯ b.
1.) How to show that ≪r is is a partial order over A/ ≃ where ≪r is the reflex cause of ≪.
2.) Is it true that A has a minimal element according to ⪯ if, and only if, A/ ≃ has a minimal element according to
≪r ? How to show if it is true or false (in such case suggest a counterexample)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Recommended textbooks for you
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,