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Say a set A is countable if there is an injective map f: A → N. Show that the product of countable sets is countable. Show that the set of quadratic integers is countable. (A number is a quadratic integer if it is the solution of a quadratic equation with integer coefficients and leading coefficient equal to 1.)

Say a set A is countable if there is an injective map f: A → N. Show that the product of countable sets is countable. Show that the set of quadratic integers is countable. (A number is a quadratic integer if it is the solution of a quadratic equation with integer coefficients and leading coefficient equal to 1.)

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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Say a set \( A \) is countable if there is an injective map \( f : A \to \mathbb{N} \). Show that the product of countable sets is countable. Show that the set of quadratic integers is countable. (A number is a quadratic integer if it is the solution of a quadratic equation with integer coefficients and leading coefficient equal to 1.)
Transcribed Image Text:Say a set \( A \) is countable if there is an injective map \( f : A \to \mathbb{N} \). Show that the product of countable sets is countable. Show that the set of quadratic integers is countable. (A number is a quadratic integer if it is the solution of a quadratic equation with integer coefficients and leading coefficient equal to 1.)
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