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A to show that the set of all open intervals with endpoints in Q union {+infty, -infty} is countable, define a function f from that collection into the Cartesian product of Q union {+infty,- infty} with itself by mapping the interval (a,b) to the ordered pair (a,b). That function is one-to-one, so the image has the same cardinality as the original collection. Since the image is a subset of that Cartesian product, which is countable, the image is countable. Prove that the collection of all intervals in R with endpoints in Qu{too) is countable.

A to show that the set of all open intervals with endpoints in Q union {+infty, -infty} is countable, define a function f from that collection into the Cartesian product of Q union {+infty,- infty} with itself by mapping the interval (a,b) to the ordered pair (a,b). That function is one-to-one, so the image has the same cardinality as the original collection. Since the image is a subset of that Cartesian product, which is countable, the image is countable. Prove that the collection of all intervals in R with endpoints in Qu{too) is countable.

Advanced Engineering Mathematics
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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Show that the set of all [a,b] with a<=b is also countable

Show that the set of all (a,b] with a<b is also countable

Show that the set of all [a,b) with a<b is also countable 

A to show that the set of all open intervals with endpoints in Q union {+infty, -infty} is countable, define a function f from that collection into the Cartesian product of Q union {+infty,- infty} with itself by mapping the interval (a,b) to the ordered pair (a,b). That function is one-to-one, so the image has the same cardinality as the original collection. Since the image is a subset of that Cartesian product, which is countable, the image is countable.
Transcribed Image Text:A to show that the set of all open intervals with endpoints in Q union {+infty, -infty} is countable, define a function f from that collection into the Cartesian product of Q union {+infty,- infty} with itself by mapping the interval (a,b) to the ordered pair (a,b). That function is one-to-one, so the image has the same cardinality as the original collection. Since the image is a subset of that Cartesian product, which is countable, the image is countable.
Prove that the collection of all intervals in R with endpoints in Qu{too) is countable.
Transcribed Image Text:Prove that the collection of all intervals in R with endpoints in Qu{too) is countable.
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