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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
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|
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Knowledge Booster
Complexity
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,