If it is possible to label each element of an infinite set S with a finite string of keyboard characters, from a finite list characters, where no two elements of Shave the same label, then S is a countably infinite set. Use the above statement and prove that the set of rational numbers is countable.
If it is possible to label each element of an infinite set S with a finite string of keyboard characters, from a finite list characters, where no two elements of Shave the same label, then S is a countably infinite set. Use the above statement and prove that the set of rational numbers is countable.
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter2: Working With Real Numbers
Section2.1: Basic Assumptions
Problem 40WE
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![If it is possible to label each element of an infinite set S with a finite string of keyboard characters, from a finite list characters, where no two elements of S have
the same label, then S is a countably infinite set.
Use the above statement and prove that the set of rational numbers is countable.
Multiple Choice
O
We can label the rational numbers with strings from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, /, -] by writing down the string that represents that rational
number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.
We can label the rational numbers with strings from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, /] by writing down the string that represents that rational
number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.
O
We can label the rational numbers with strings from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -] by writing down the string that represents that rational
number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.
O
We can label the rational numbers with strings from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, /, -] by writing down the string that represents that rational
number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bdd3ed4-f5c8-4322-afa4-78a84ffe811c%2F4ee96c0f-41ab-424a-b947-da7dab864cf3%2Fxdsflrgt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:If it is possible to label each element of an infinite set S with a finite string of keyboard characters, from a finite list characters, where no two elements of S have
the same label, then S is a countably infinite set.
Use the above statement and prove that the set of rational numbers is countable.
Multiple Choice
O
We can label the rational numbers with strings from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, /, -] by writing down the string that represents that rational
number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.
We can label the rational numbers with strings from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, /] by writing down the string that represents that rational
number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.
O
We can label the rational numbers with strings from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -] by writing down the string that represents that rational
number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.
O
We can label the rational numbers with strings from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, /, -] by writing down the string that represents that rational
number in its simplest form. As the labels are unique, it follows that the set of rational numbers is countable.
![The set A x Z* where, A = (2, 3) (Check all that apply.)
Check All That Apply
The set is countable.
The set is countably infinite with one-to-one correspondence 1 → (2,1), 2 ↔ (3,1), 3 ↔ (2,2), 4 ↔ (3,2),and so on.
The set is countably infinite with one-to-one correspondence 0 ↔ (2,1),1 ↔ (3,1), 2 ↔ (2,2), 3 → (3,2), and so on.
The set is uncountable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bdd3ed4-f5c8-4322-afa4-78a84ffe811c%2F4ee96c0f-41ab-424a-b947-da7dab864cf3%2F2oxv3i9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The set A x Z* where, A = (2, 3) (Check all that apply.)
Check All That Apply
The set is countable.
The set is countably infinite with one-to-one correspondence 1 → (2,1), 2 ↔ (3,1), 3 ↔ (2,2), 4 ↔ (3,2),and so on.
The set is countably infinite with one-to-one correspondence 0 ↔ (2,1),1 ↔ (3,1), 2 ↔ (2,2), 3 → (3,2), and so on.
The set is uncountable.
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