,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
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Just a and b
b) Use the idea in part (a) to determine an explicit one-to-one correspondence between
zt and the set of all ordered pairs of positive integers.
c) Use part (a) to give a different proof of Theorem 1.26.
Transcribed Image Text:b) Use the idea in part (a) to determine an explicit one-to-one correspondence between zt and the set of all ordered pairs of positive integers. c) Use part (a) to give a different proof of Theorem 1.26.
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), ....
a) Show that this constitutes a proof that the set of all ordered pairs of positive integers
is countably infinite.
Transcribed Image Text: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), .... a) Show that this constitutes a proof that the set of all ordered pairs of positive integers is countably infinite.
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