Question 1. Let A and B be sets. (a) Prove that P(A) nP(B) = P(ANB). (b) Prove that P(A) UP(B) ≤ P(AUB). (c) Give an example of sets A, B such that P(A) UP(B) ‡ P(AUB). Question 2. Suppose that A, B and C are sets which satisfy both of the following conditions: (a) AUC BUC; (b) AnCBN C. Prove that A = B. Question 3. Let = be the relation defined on wxw by (a, b) = (c, d) ad = cb. Determine whether or not = is an equivalence relation on wxw. (Of course, it is not enough merely to state that = is an equivalence relation or to state that = is not an equivalence relation. You must justify your answer!) Question 4. Let h: ww be the function defined recursively by h(0): = 1 h(n + 1) = 5h(n) + 7n. Prove by induction that for all new, h(n) = 5n+7n 2
Question 1. Let A and B be sets. (a) Prove that P(A) nP(B) = P(ANB). (b) Prove that P(A) UP(B) ≤ P(AUB). (c) Give an example of sets A, B such that P(A) UP(B) ‡ P(AUB). Question 2. Suppose that A, B and C are sets which satisfy both of the following conditions: (a) AUC BUC; (b) AnCBN C. Prove that A = B. Question 3. Let = be the relation defined on wxw by (a, b) = (c, d) ad = cb. Determine whether or not = is an equivalence relation on wxw. (Of course, it is not enough merely to state that = is an equivalence relation or to state that = is not an equivalence relation. You must justify your answer!) Question 4. Let h: ww be the function defined recursively by h(0): = 1 h(n + 1) = 5h(n) + 7n. Prove by induction that for all new, h(n) = 5n+7n 2
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...
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Transcribed Image Text:Question 1. Let A and B be sets.
(a) Prove that P(A) nP(B) = P(ANB).
(b) Prove that P(A) UP(B) ≤ P(AUB).
(c) Give an example of sets A, B such that P(A) UP(B) ‡ P(AUB).
Question 2. Suppose that A, B and C are sets which satisfy both of the following
conditions:
(a) AUC
BUC;
(b) AnCBN C.
Prove that A = B.
Question 3. Let = be the relation defined on wxw by
(a, b) = (c, d)
ad = cb.
Determine whether or not = is an equivalence relation on wxw.
(Of course, it is not enough merely to state that = is an equivalence relation or to
state that = is not an equivalence relation. You must justify your answer!)
Question 4. Let h: ww be the function defined recursively by
h(0):
=
1
h(n + 1) = 5h(n) + 7n.
Prove by induction that for all new,
h(n) =
5n+7n
2
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