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MT1101 Three hours The total number of marks on the paper is 80. A further 20 marks are available from work during the semester making a total of 100. UNIVERSITY OF MANCHESTER SETS, NUMBERS AND FUNCTIONS: SAMPLE EXAM Someday Sometime Answer ALL SIX questions in Section A (26 marks in all) and THREE of the five questions in Section B (54 marks in all). Electronic calculators may be used, provided that they cannot store text. 1 of 3 P.T.O. MT1101 SECTION A Answer ALL SIX questions A1. Construct truth tables for the statements: 1. P ⇐ Q 2. (not P ) or (not Q) 3. P and (not Q) 4. not (P or Q). [4 marks] A2. Prove or disprove each of the following statements: 1. ∃y ∈ R, ∀x ∈ R, x + y = 0 2. ∀x ∈ R, ∃y ∈ R, x + y = 0 3. ∀x ∈ R, ∃y ∈ R, xy = 1 4. ∃y ∈ R, ∀x ∈ R, xy = 1 5. ∃y ∈ R, ∀x ∈ R, xy = 0. [5 marks] A3. Explain what is meant by: 1. the contrapositive of a statement P ⇒ Q 2. the image of a function 3. an irrational number 4. an uncountable set. [4 marks] A4. [4 marks] A5. [4 marks] A6. [4 marks] 2 of 3 P.T.O. MT1101 SECTION B Answer THREE of the five questions B1. Describe the induction principle for statements P (n), n ∈ Z+ . Explain how the integers 2n and n! are defined inductively. Prove by induction on n that Pn 1 (1) j=1 j = 2 n(n + 1) (2) 3 divides 4n + 5 for all positive integers n. [18 marks] B2. For any function f : X → Y , define the domain and codomain of f . Describe the properties required for f to be (i) an injection, (ii) a surjection, (iii) a bijection. Explain what is meant by an inverse of f , and prove that f is invertible if and only if it is a bijection. Let h: R → R be defined by h(x) = x2 , ∀x ∈ R, and show that h is neither an injection nor a surjection. Explain how the domain and codomain may be restricted so that h becomes (i) an injection but not a surjection, (ii) a surjection but not an injection, (iii) a bijection. In case (iii), give an explicit description of h−1 . [18 marks] B3. (i) Explain what is meant by a denumerable set, and prove that Z is denumerable. Give two examples (without proof) of infinite sets which are not denumerable. Give a description of Z as the union of (1) a finite set and a denumerable set (2) three denumerable sets. [9 marks] (ii) [9 marks] B4. [18 marks] B5. [18 marks] END OF EXAMINATION PAPER 3 of 3