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Suppose that B is a subset of A. Specify a one-to-one function whose domain is B and whose range is a subset of A.By doing this you will show that |B| < |A||

Suppose that B is a subset of A. Specify a one-to-one function whose domain is B and whose range is a subset of A.By doing this you will show that |B| < |A||

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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Note. If B is a subset of A and A is countable then B is countable. To see this let a1, a2 . . . be a sequence whose range is A. This means that every agis a member of A and every member of A is ar for some k . Then define the sequence b1, b2, ... in the following way. Let ji be the first integer j such that a, e B. Given that jr has been found, let jk+1 be the first integer j > jk such that a; e B. Do this for k = 1, 2, .... Let br %3D ajr for k = 1, 2,.... The sequence b1, b2 . .. will have range B. %3! Note. According to the definition, when each of A and B is a set, |B| < |A| means that there is a one-to-one function from B into A. Suppose that B is a subset of A. Specify a one-to-one function whose domain is B and whose range is a subset of A.By doing this you will show that |B| < |A|
Transcribed Image Text:Note. If B is a subset of A and A is countable then B is countable. To see this let a1, a2 . . . be a sequence whose range is A. This means that every agis a member of A and every member of A is ar for some k . Then define the sequence b1, b2, ... in the following way. Let ji be the first integer j such that a, e B. Given that jr has been found, let jk+1 be the first integer j > jk such that a; e B. Do this for k = 1, 2, .... Let br %3D ajr for k = 1, 2,.... The sequence b1, b2 . .. will have range B. %3! Note. According to the definition, when each of A and B is a set, |B| < |A| means that there is a one-to-one function from B into A. Suppose that B is a subset of A. Specify a one-to-one function whose domain is B and whose range is a subset of A.By doing this you will show that |B| < |A|
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