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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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