Problem Set 1 - Introduction to Proof | MATH 307, Assignments of Mathematics

Material Type: Assignment; Class: Introduction to Proof >4; Subject: Mathematics; University: University of Oregon; Term: Unknown 1989;

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Pre 2010

Uploaded on 09/17/2009

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Math 307
Final Homework, not to be turned it
1. Let an=(โˆ’1)n
n+1 . Prove, from the definitions, that the sequence aconverges to 0.
2. Let an=3n2
n2+1 . Prove, from the definitions, that aconverges to 3.
3. Let an=n3+ 1. Prove, from the definitions, that the sequence adoes not converge to any real number.
4. Let f:Rโ†’Rbe given by f(x) = 2xโˆ’5. Prove, from the definitions, that fis continuous at 1.
5. A function f:Sโ†’Tis said to be onto if (โˆ€xโˆˆT)(โˆƒaโˆˆS)[f(a) = x].
(a) Let f:Z5โ†’Z5be given by f(x) = x2. Is this function onto?
(b) Let f:Z7โ†’Z7be given by f(x) = x3. Is this function onto?
(c) Suppose given that f:Sโ†’Tis onto, and that AโІT. Prove that f(If(A)) = A.

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Math 307 Final Homework, not to be turned it

  1. Let an = (โˆ’1) n n+1. Prove, from the definitions, that the sequence^ a^ converges to 0.
  2. Let an = (^) n^32 n+1^2. Prove, from the definitions, that a converges to 3.
  3. Let an = n^3 + 1. Prove, from the definitions, that the sequence a does not converge to any real number.
  4. Let f : R โ†’ R be given by f (x) = 2x โˆ’ 5. Prove, from the definitions, that f is continuous at 1.
  5. A function f : S โ†’ T is said to be onto if (โˆ€x โˆˆ T )(โˆƒa โˆˆ S)[f (a) = x]. (a) Let f : Z 5 โ†’ Z 5 be given by f (x) = x^2. Is this function onto? (b) Let f : Z 7 โ†’ Z 7 be given by f (x) = x^3. Is this function onto? (c) Suppose given that f : S โ†’ T is onto, and that A โІ T. Prove that f (If (A)) = A.