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Let SCR be the set of algebraic numbers. Prove that S is countable. Hint: You may use the fact (which you might have seen in class or you can prove separa- if I is a countable set and for each i, A, is a countable set then Uie1 A; is a countable set Hint: You may also use the fact that a polynomial of degree n can have at most n real For Q4-5, suppose that A, B CR are nonempty and bounded from above. Prove that if ACB, then sup(A) ≤ sup(B). Define A + B = {a+b: a € A, b = B}. a) Determine {1,3,5} + {-3,0, 1}. b) Prove that sup(A + B) = sup(A) + sup(B). It is not easy to prove that a number is transcendental.

Let SCR be the set of algebraic numbers. Prove that S is countable. Hint: You may use the fact (which you might have seen in class or you can prove separa- if I is a countable set and for each i, A, is a countable set then Uie1 A; is a countable set Hint: You may also use the fact that a polynomial of degree n can have at most n real For Q4-5, suppose that A, B CR are nonempty and bounded from above. Prove that if ACB, then sup(A) ≤ sup(B). Define A + B = {a+b: a € A, b = B}. a) Determine {1,3,5} + {-3,0, 1}. b) Prove that sup(A + B) = sup(A) + sup(B). It is not easy to prove that a number is transcendental.

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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Let SCR be the set of algebraic numbers. Prove that S is countable. Hint: You may use the fact (which you might have seen in class or you can prove separa if I is a countable set and for each i, A; is a countable set then Uiel Ai is a countable set. Hint: You may also use the fact that a polynomial of degree n can have at most n real For Q4-5, suppose that A, B CR are nonempty and bounded from above. Prove that if ACB, then sup(A) ≤ sup(B). Define A + B = {a+b: a EA, b E B}. a) Determine {1, 3, 5} + {-3,0, 1}. b) Prove that sup (A + B) = sup(A) + sup(B). It is not easy to prove that a number is transcendental.
Transcribed Image Text:Let SCR be the set of algebraic numbers. Prove that S is countable. Hint: You may use the fact (which you might have seen in class or you can prove separa if I is a countable set and for each i, A; is a countable set then Uiel Ai is a countable set. Hint: You may also use the fact that a polynomial of degree n can have at most n real For Q4-5, suppose that A, B CR are nonempty and bounded from above. Prove that if ACB, then sup(A) ≤ sup(B). Define A + B = {a+b: a EA, b E B}. a) Determine {1, 3, 5} + {-3,0, 1}. b) Prove that sup (A + B) = sup(A) + sup(B). It is not easy to prove that a number is transcendental.
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