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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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.
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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