I need help with the question below. (Question 5 section 6.1 is attached below)Recall the relation in Exercise 5 of Section 6.1, ρ defined on the power set, P(S), of a set S. The definition was (A,B) ∈ ρ iff A∩B=∅. Draw the digraph for ρ where S={a,b}. 5. Let p be the relation on the power set, P(S), of a finite set S of cardinality ndefined p by (A, B) e p iff An B = 0.a. Consider the specific case n = 3, and determine the cardinality of the setp.b. What is the cardinality of p for an arbitrary n? Express your answer interms of n. (Hint: There are three places that each element of S can go inbuilding an element of p.)Answera. When n = 3, there are 27 pairs in the relation.b. Imagine building a pair of disjoint subsets of S. For each element of Sthere are three places that it can go: into the first set of the orderedpair, into the second set, or into neither set. Therefore the number ofpairs in the relation is 3", by the product rule.

Here, S = a,b Therefore, P(S) = ϕ, a, b, SNow, A,B ∈ l iff A ∩ B = ϕSo, ϕ ∩ a = ϕ, ϕ ∩ b = ϕ , ϕ ∩ S = ϕ, ϕ ∩ ϕ = ϕ , a ∩ b = ϕ, b ∩ a = ϕTherefore,l = ϕ,ϕ, ϕ,b, ϕ,S, ϕ,a, a, b, b, a .Therefore diagraph of the relation P is,

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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I need help with the question below. (Question 5 section 6.1 is attached below)

Recall the relation in Exercise 5 of Section 6.1, ρ defined on the power set, P(S), of a set S. The definition was (A,B) ∈ ρ iff A∩B=∅. Draw the digraph for ρ where S={a,b}.

5. Let p be the relation on the power set, P(S), of a finite set S of cardinality n
defined p by (A, B) e p iff An B = 0.
a. Consider the specific case n = 3, and determine the cardinality of the set
p.
b. What is the cardinality of p for an arbitrary n? Express your answer in
terms of n. (Hint: There are three places that each element of S can go in
building an element of p.)
Answer
a. When n = 3, there are 27 pairs in the relation.
b. Imagine building a pair of disjoint subsets of S. For each element of S
there are three places that it can go: into the first set of the ordered
pair, into the second set, or into neither set. Therefore the number of
pairs in the relation is 3", by the product rule.

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