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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.
Typology: Exams
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Instructors: Omer Angel, Dragos Ghioca
First Name: Last Name:
Student id. Section (circle): 201 202
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 โ A and D โ B. Define the sets f โ^1 (D)
(b) Define the supremum of a set S of real numbers.
(c) State the converse of the statement โIf I will win the lottery then I will buy a new car.โ
(d) Write the negation of the statement โx โ R, โy โ 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 โNoneโ. You do not need to prove your answers. R write its supremum and infimum if they exist. If
(a)^ { x โ R s.t. โ 1 < x < 5 } (b)^ { x โ Q s.t. 3 โค x^2 โค 7 } (c) nโ^ โ=
2 + n^1 , 6 โ (^) n^2
(d) nโ^ โ=
n, n
(10 marks) 3. Using the definition of convergence for sequences, prove that (a) (^) nlimโโ^ n^2 + 3 n 2 n + 1= 1. (b) (^) nlimโโ^1 โ^ 2 cos( n n)= 0.
(5 marks) 5. Prove that the function f : R โ { 1 } โ R โ { 2 } given by f (x) = (^) x^2 โx 1 is bijective.
(5 marks) 6. Let n โ Z. Prove that n โก 3 (mod 5) if and only if 3n + 1 is divisible by 5.
(12 marks) 8. Letfollowing statements are true or false. P โ N be the set of prime numbers Prove your answers (โtrueโ or โfalseโ is not P = { 2 , 3 , 5 , 7 ,... }. Determine whether the sufficient). (a) โm โ P, โn โ P, m + n โ P. (b) โm โ P, โn โ P s.t. m + n โ P. (c) โm โ P s.t. โn โ P, m + n โ P. (d) โm โ P s.t. โn โ P s.t. m + n โ P.
(15 marks) 9. Let f : A โ B and g : B โ A be functions so that g โฆ 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 โฆ g must be surjective.
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