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Problem 2.1 Suppose that a number x is to be selected from the real line R, and let A, B, and C be the events represented by the following subsets of R, where the notation {x : - --} denotes the set containing every point for which the property presented following the colon is satisfied: {x :1

Problem 2.1 Suppose that a number x is to be selected from the real line R, and let A, B, and C be the events represented by the following subsets of R, where the notation {x : - --} denotes the set containing every point for which the property presented following the colon is satisfied: {x :1

Advanced Engineering Mathematics
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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Problem 2.1 Suppose that a number x is to be selected from the real line R, and let A, B, and C be the events represented by the following subsets of R, where the notation {x : - --} denotes the set containing every point x for which the property presented following the colon is satisfied: {x :1<x < 5} {x : 3 < x < 7} {x : x < 0} A В C Describe each of the following events as a set of real numbers: (а) Ас (b) AUB (c) BnCc (d) Aºn Bº n Cc (e) (AU B) NC Problem 2.2 Toss a coin 4 times. Let A denote the event that a head is obtained on the first toss, and let B denote the event that a head is obtained on the fourth toss. Is AN B empty? Problem 2.3 Let A and B be two sets. (a) Show that (A°n B®)° = AUB and (A° U B°)° = An B. (b) Consider rolling a six-sided die once. Let A be the set of outcomes where an odd number comes up. Let B be the set of outcomes where a 1 or a 2 comes up. Calculate the sets on both sides of the equalities in part (a), and verify that the equalities hold. Problem 2.4 Let A and B be two sets with a finite number of elements. Show that the number of elements in ANB plus the number of elements in AUB is equal to the number of elements in A plus the number of elements in B. Problem 2.5 Show that if A and Bn (n = 1,2, 3, ...) are events, then An (U Bn) = U (AN B,)
Transcribed Image Text:Problem 2.1 Suppose that a number x is to be selected from the real line R, and let A, B, and C be the events represented by the following subsets of R, where the notation {x : - --} denotes the set containing every point x for which the property presented following the colon is satisfied: {x :1<x < 5} {x : 3 < x < 7} {x : x < 0} A В C Describe each of the following events as a set of real numbers: (а) Ас (b) AUB (c) BnCc (d) Aºn Bº n Cc (e) (AU B) NC Problem 2.2 Toss a coin 4 times. Let A denote the event that a head is obtained on the first toss, and let B denote the event that a head is obtained on the fourth toss. Is AN B empty? Problem 2.3 Let A and B be two sets. (a) Show that (A°n B®)° = AUB and (A° U B°)° = An B. (b) Consider rolling a six-sided die once. Let A be the set of outcomes where an odd number comes up. Let B be the set of outcomes where a 1 or a 2 comes up. Calculate the sets on both sides of the equalities in part (a), and verify that the equalities hold. Problem 2.4 Let A and B be two sets with a finite number of elements. Show that the number of elements in ANB plus the number of elements in AUB is equal to the number of elements in A plus the number of elements in B. Problem 2.5 Show that if A and Bn (n = 1,2, 3, ...) are events, then An (U Bn) = U (AN B,)
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