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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.

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.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 20E: Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not...
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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: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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