Math 220 Final Exam - University of British Columbia, April 2011, Exams of Mathematics

The instructions and questions for the final exam of math 220 - introduction to mathematical proof, held at the university of british columbia in april 2011. The exam covers various topics in mathematical proofs, sequences, series, set theory, and logic.

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The University of British Columbia
Math 220 โ€” Introduction to Mathematical Proof
2011, April 21
Instructors: Omer Angel, Dragos Ghioca
First Name: Last Name:
Student id. Section (circle): 201 202
Instructions
โ€ขThis exam consists of 10 questions worth a total of 100 points.
โ€ขJustify all answers, show all work and calculations. Extra paper is available as needed.
โ€ขNo calculators or other aids are permitted.
โ€ขMake sure this exam has 12 pages excluding this cover page.
โ€ขDuration: 2 hours 30 minutes.
Good luck, and enjoy your summer.
1. Each candidate should be prepared to produce his library/AMS card
upon request.
2. Read and observe the following rules:
No candidate shall be permitted to enter the examination room after the expiration
of one half hour, or to leave during the first half hour of the examination.
Candidates are not permitted to ask questions of the invigilators, except in cases of
supposed errors or ambiguities in examination questions.
CAUTION - Candidates guilty of any of the following or similar practices shall
be immediately dismissed from the examination and shall be liable to disciplinary
action.
(a) Making use of any books, papers or memoranda, other than those authorized by
the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates. The plea of
accident or forgetfulness shall not be received.
3. Smoking is not permitted during examinations.
Question: 1 2 3 4 5 6 7 8 9 10 Total
Points: 10 8 10 5 5 5 15 12 15 15 100
Score:
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The University of British Columbia

Math 220 โ€” Introduction to Mathematical Proof

2011, April 21

Instructors: Omer Angel, Dragos Ghioca

First Name: Last Name:

Student id. Section (circle): 201 202

Instructions

  • โ€ข This exam consists ofJustify all answers, show all work and calculations. Extra paper is available as needed. 10 questions worth a total of 100 points.
  • โ€ข No calculators or other aids are permitted.Make sure this exam has 12 pages excluding this cover page.
  • Duration: 2 hours 30 minutes. Good luck, and enjoy your summer.
    1. Each candidate should be prepared to produce his library/AMS cardupon request.
    2. Read and observe the following rulesNo candidate shall be permitted to enter the examination room after the expiration: of one half hour, or to leave during the first half hour of the examination.Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinaryaction. (a) Making use of any books, papers or memoranda, other than those authorized bythe examiners. (b) Speaking or communicating with other candidates.(c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received. 3. Smoking is not permitted during examinations.

Question: 1 2 3 4 5 6 7 8 9 10 Total Points: 10 8 10 5 5 5 15 12 15 15 100 Score:

(10 marks) 1. (a) Letand ff (^) (:C A). โ†’ B be a function and let C โІ A and D โІ B. Define the sets f โˆ’^1 (D)

(b) Define the supremum of a set S of real numbers.

(c) State the converse of the statement โ€œIf I will win the lottery then I will buy a new car.โ€

(d) Write the negation of the statement โˆ€x โˆˆ R, โˆƒy โˆˆ N s.t. x^2 + y < x + y^2.

(e) Define what it means for a function f : A โ†’ B to be injective.

(8 marks) 2. For each of the following subsets ofthey do not exist write โ€œNoneโ€. You do not need to prove your answers. R write its supremum and infimum if they exist. If

(a)^ { x โˆˆ R s.t. โˆ’ 1 < x < 5 } (b)^ { x โˆˆ Q s.t. 3 โ‰ค x^2 โ‰ค 7 } (c) nโ‹‚^ โˆž=

[

2 + n^1 , 6 โˆ’ (^) n^2

]

(d) nโ‹ƒ^ โˆž=

[ 1

n, n

]

(10 marks) 3. Using the definition of convergence for sequences, prove that (a) (^) nlimโ†’โˆž^ n^2 + 3 n 2 n + 1= 1. (b) (^) nlimโ†’โˆž^1 โˆ’^ 2 cos( n n)= 0.

(5 marks) 5. Prove that the function f : R โˆ’ { 1 } โ†’ R โˆ’ { 2 } given by f (x) = (^) x^2 โˆ’x 1 is bijective.

(5 marks) 6. Let n โˆˆ Z. Prove that n โ‰ก 3 (mod 5) if and only if 3n + 1 is divisible by 5.

(12 marks) 8. Letfollowing statements are true or false. P โŠ‚ N be the set of prime numbers Prove your answers (โ€œtrueโ€ or โ€œfalseโ€ is not P = { 2 , 3 , 5 , 7 ,... }. Determine whether the sufficient). (a) โˆ€m โˆˆ P, โˆ€n โˆˆ P, m + n โˆˆ P. (b) โˆ€m โˆˆ P, โˆƒn โˆˆ P s.t. m + n โˆˆ P. (c) โˆƒm โˆˆ P s.t. โˆ€n โˆˆ P, m + n โˆˆ P. (d) โˆƒm โˆˆ P s.t. โˆƒn โˆˆ P s.t. m + n โˆˆ P.

(15 marks) 9. Let f : A โ†’ B and g : B โ†’ A be functions so that g โ—ฆ f is a surjective function. (a) Prove that g is surjective. (b) Give an example of sets A, B and functions f, g as above such that f is not surjective. (c) Prove or disprove that f โ—ฆ g must be surjective.

Question: