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Suppose that a number z 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 {r : ---} denotes the set containing every point z for which the property presented following the colon is satisfied: A = {r:1

Suppose that a number z 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 {r : ---} denotes the set containing every point z for which the property presented following the colon is satisfied: A = {r: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 \( \mathbb{R} \), and let \( A, B, \) and \( C \) be the events represented by the following subsets of \( \mathbb{R} \), where the notation \(\{x : - - -\}\) denotes the set containing every point \( x \) for which the property presented following the colon is satisfied: - \( A = \{x: 1 \leq x \leq 5\} \) - \( B = \{x: 3 < x \leq 7\} \) - \( C = \{x: x \leq 0\} \) Describe each of the following events as a set of real numbers: (a) \( A^c \) (b) \( A \cup B \) (c) \( B \cap C^c \) (d) \( A^c \cap B^c \cap C^c \) (e) \( (A \cup B) \cap C \) **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 \( A \cap B \) empty? **Problem 2.3** Let \( A \) and \( B \) be two sets. (a) Show that \( (A^c \cap B^c)^c = A \cup B \) and \( (A^c \cup B^c)^c = A \cap B \). (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 \( A \cap B \) plus the number of elements in \( A \cup B \) is equal to the number of elements in \( A
Transcribed Image Text:**Problem 2.1** Suppose that a number \( x \) is to be selected from the real line \( \mathbb{R} \), and let \( A, B, \) and \( C \) be the events represented by the following subsets of \( \mathbb{R} \), where the notation \(\{x : - - -\}\) denotes the set containing every point \( x \) for which the property presented following the colon is satisfied: - \( A = \{x: 1 \leq x \leq 5\} \) - \( B = \{x: 3 < x \leq 7\} \) - \( C = \{x: x \leq 0\} \) Describe each of the following events as a set of real numbers: (a) \( A^c \) (b) \( A \cup B \) (c) \( B \cap C^c \) (d) \( A^c \cap B^c \cap C^c \) (e) \( (A \cup B) \cap C \) **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 \( A \cap B \) empty? **Problem 2.3** Let \( A \) and \( B \) be two sets. (a) Show that \( (A^c \cap B^c)^c = A \cup B \) and \( (A^c \cup B^c)^c = A \cap B \). (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 \( A \cap B \) plus the number of elements in \( A \cup B \) is equal to the number of elements in \( A
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