Let A = {1, 2, 3} and let ρ be a relation on P (A) × Z that maps a set to its cardinality, e.g. ({1, 2}, 2) ∈ ρ. (The notation P (A) denotes the powerset of A) What is the set B ⊂ Z with the fewest elements such that ρ ⊂ P (A) × B? Provide a written justification as well as the set B itself.
Let A = {1, 2, 3} and let ρ be a relation on P (A) × Z that maps a set to its cardinality, e.g. ({1, 2}, 2) ∈ ρ. (The notation P (A) denotes the powerset of A) What is the set B ⊂ Z with the fewest elements such that ρ ⊂ P (A) × B? Provide a written justification as well as the set B itself.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 5TFE: True or False
Label each of the following statements as either true or false.
Let be an equivalence...
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Question
i. Let A = {1, 2, 3} and let ρ be a relation on P (A) × Z that maps a set to its cardinality, e.g.
({1, 2}, 2) ∈ ρ. (The notation P (A) denotes the powerset of A)
What is the set B ⊂ Z with the fewest elements such that ρ ⊂ P (A) × B? Provide a written
justification as well as the set B itself.
ii. For two non-empty sets E, F , explain whether or not it is possible that both E ∈ F and
E ⊆ F .
iii. For two non-empty sets E, F , explain whether or not it is possible that both E ∈ P (F ) and
E ⊆ P (F )
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