(6) A subset A of R is called co-finite if its complement R\ A is finite. What is the cardinality of the set of all co-finite subsets of R?

Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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(5) Let S be the circle (x – a)² + (y – b)² = r2 where a, b, r are all constructible numbers.
Let S, be the set of those points on S both of whose coordinates are constructible and
S2 be the set of those points on S whose at least one coordinate is not constructible.
Show that |S1| < |S2\.
You may use the fact that |S| = |R| = c.
(6) A subset A of R is called co-finite if its complement R \ A is finite. What is the
cardinality of the set of all co-finite subsets of R?
(7) Let a E R be an algebraic number. The minimal polynomial of a, denoted by pa(x),
Transcribed Image Text:(5) Let S be the circle (x – a)² + (y – b)² = r2 where a, b, r are all constructible numbers. Let S, be the set of those points on S both of whose coordinates are constructible and S2 be the set of those points on S whose at least one coordinate is not constructible. Show that |S1| < |S2\. You may use the fact that |S| = |R| = c. (6) A subset A of R is called co-finite if its complement R \ A is finite. What is the cardinality of the set of all co-finite subsets of R? (7) Let a E R be an algebraic number. The minimal polynomial of a, denoted by pa(x),
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