,1), (2, 2), (1, 3), (4, 1), (3, 2), (2, 3), (1,4), (5, 1), .... Show that this constitutes a proof that the set of all ordered pairs of positive integers is countably infinite.
,1), (2, 2), (1, 3), (4, 1), (3, 2), (2, 3), (1,4), (5, 1), .... Show that this constitutes a proof that the set of all ordered pairs of positive integers is countably infinite.
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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Question
Just a and b
Transcribed Image Text:The image contains the following text:
"b) Use the idea in part (a) to determine an explicit one-to-one correspondence between ℤ⁺ and the set of all ordered pairs of positive integers.
c) Use part (a) to give a different proof of Theorem 1.26."
Note: There are no graphs or diagrams in the image.
Transcribed Image Text:**Title: Countability of Ordered Pairs of Positive Integers**
**Description:**
Explore how the set of all ordered pairs of positive integers can be systematically listed and proven as countably infinite.
**Content:**
Interpret the set of all ordered pairs of positive integers as a grid of dots in the first quadrant of the xy-plane. Consider the "path" that traverses these dots in the following order:
(1, 1), (2, 1), (1, 2), (3, 1), (2, 2), (1, 3), (4, 1), (3, 2), (2, 3), (1, 4), (5, 1), ...
**Task:**
a) Show that this constitutes a proof that the set of all ordered pairs of positive integers is countably infinite.
**Explanation:**
By following this path, every ordered pair is visited without repetition. This systematic listing shows a one-to-one correspondence between the ordered pairs and natural numbers, demonstrating that the set is countably infinite.
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