Consider the following argument for showing that |R| = |P(N)|. By the Cantor-Bernstein-Schröder theorem (page 17, following Definition 0.3.28), it suffices to find an injection from P(N) into R and also an injection from R into P(N). When S is a set of natural numbers, create a real number is in the following way. Start from the decimal 0.123456789101112... obtained by writing down the natural numbers in their standard order, and for each element n of the set S, replace the instance of ʼn in this decimal by a string of zeros (one for each digit of n). For example, if S is the set of even natural numbers, then as is the decimal 0.10305070900110013.... The function S→ as is an injection from P(N) into R. According to the discussion on page 38 (after Proposition 1.4.1), all intervals have the same cardinality, so instead of producing an injection from R into P(N), it suffices to find an injection from (0, 1) into P(N). When a is a real number between 0 and 1, write a as a nonterminating decimal. Associate to x a set S₂ of natural numbers in the following way. For each positive integer n, if the nth decimal digit of x is not 0, form a natural number by appending n zeros to this digit, and put this natural number into the set S. For example, if x = 0.409 ..., then S₂ contains the natural numbers 40 and 9000 and so forth. The function → S₂ is an injection from (0, 1) into P(N). True or false: This argument proves the claim on page 39 just above Theorem 1.4.2 that |R| = |P(N)|.
Consider the following argument for showing that |R| = |P(N)|. By the Cantor-Bernstein-Schröder theorem (page 17, following Definition 0.3.28), it suffices to find an injection from P(N) into R and also an injection from R into P(N). When S is a set of natural numbers, create a real number is in the following way. Start from the decimal 0.123456789101112... obtained by writing down the natural numbers in their standard order, and for each element n of the set S, replace the instance of ʼn in this decimal by a string of zeros (one for each digit of n). For example, if S is the set of even natural numbers, then as is the decimal 0.10305070900110013.... The function S→ as is an injection from P(N) into R. According to the discussion on page 38 (after Proposition 1.4.1), all intervals have the same cardinality, so instead of producing an injection from R into P(N), it suffices to find an injection from (0, 1) into P(N). When a is a real number between 0 and 1, write a as a nonterminating decimal. Associate to x a set S₂ of natural numbers in the following way. For each positive integer n, if the nth decimal digit of x is not 0, form a natural number by appending n zeros to this digit, and put this natural number into the set S. For example, if x = 0.409 ..., then S₂ contains the natural numbers 40 and 9000 and so forth. The function → S₂ is an injection from (0, 1) into P(N). True or false: This argument proves the claim on page 39 just above Theorem 1.4.2 that |R| = |P(N)|.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.4: Zeros Of A Polynomial
Problem 18E: Show that the converse of Eisenstein’s Irreducibility Criterion is not true by finding an...
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As per the question, we will be verifying the argument .
Let us summarize the given information.
By the Cantor-Bernstein-Schroder theorem, it suffices to find an injection from into and also an injection from into .
Let us assume is a set of natural numbers.
So, .
Now, is created in the following way.
Start from the decimal obtained by writing down the natural numbers in their standard order, and for each element of the set , replace the instance of in this decimal by a string of zeros (one for each digit of ).
For example, if S is the set of even natural numbers, then is the decimal .
The function is an injection from into .
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