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2. Recall that for two sets A, B, the Cartesian product of A and B is the set A × B = {(a, b) : a E A and be B}. This can be extended naturally to a finite number of sets A1, A2,.…., Aµ: A1 x Az x ...x Ag = {(a1,a2,...,at) : a; €e A; for 1

2. Recall that for two sets A, B, the Cartesian product of A and B is the set A × B = {(a, b) : a E A and be B}. This can be extended naturally to a finite number of sets A1, A2,.…., Aµ: A1 x Az x ...x Ag = {(a1,a2,...,at) : a; €e A; for 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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2. Recall that for two sets A, B, the Cartesian product of A and B is the set Ax B = {(a, b) : a € A and b e B}. This can be extended naturally to a finite number of sets A1, A2,…., Aµ: Aj x Az x ...x Ag = {(a1, a2, …..,ak) : a; € A; for 1< j< k}. This can be further extended to an infinite number of sets. Let I be an index set and X = {X;}ie1 be a family of sets indexed by the set I. Note that I may be an uncountable set. The Cartesian product of the family X is given by %3D Įx = {s:1 →UX : f(1) € x,for ll 1 i! That is, the elements of the Cartesian product are functions whose domain is the index set I and the image of i is in X¡ for any i e I. Let {G¡}ie1 be a family of groups indexed by the set I. Let e; be the identity element of G¡, for all i e I. Consider the Cartesian product G = [IIkes Gi- c. Let {p;}1 be the set of all the prime numbers. Show that any rational number q E Q can be written in the form q = E where n; € Z+ for all j € Z+, and all but finitely many of the x;'s are 0. Note that this representation is NOT unique.
Transcribed Image Text:2. Recall that for two sets A, B, the Cartesian product of A and B is the set Ax B = {(a, b) : a € A and b e B}. This can be extended naturally to a finite number of sets A1, A2,…., Aµ: Aj x Az x ...x Ag = {(a1, a2, …..,ak) : a; € A; for 1< j< k}. This can be further extended to an infinite number of sets. Let I be an index set and X = {X;}ie1 be a family of sets indexed by the set I. Note that I may be an uncountable set. The Cartesian product of the family X is given by %3D Įx = {s:1 →UX : f(1) € x,for ll 1 i! That is, the elements of the Cartesian product are functions whose domain is the index set I and the image of i is in X¡ for any i e I. Let {G¡}ie1 be a family of groups indexed by the set I. Let e; be the identity element of G¡, for all i e I. Consider the Cartesian product G = [IIkes Gi- c. Let {p;}1 be the set of all the prime numbers. Show that any rational number q E Q can be written in the form q = E where n; € Z+ for all j € Z+, and all but finitely many of the x;'s are 0. Note that this representation is NOT unique.
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