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Three hours The number of marks available on this paper is 75. THE UNIVERSITY OF MANCHESTER SETS, NUMBERS AND FUNCTIONS ?? January 2015 ??.?? – ??.?? Answer ALL FIVE questions in Section A (30 marks in total). Answer THREE of the FIVE questions in Section B (45 marks in all). If more than THREE questions from Section B are attempted, then credit will be given for the best THREE answers. Electronic calculators may be used provided they cannot store text. 1 SECTION A 2 Answer ALL FIVE questions A1. Construct truth tables for the statements: (i) (ii) (iii) (iv) P ⇒Q P and (not Q) (not P ) or (not Q) (P or Q) ⇐ R. [6 marks] A2. Prove or disprove each of the following statements: (i) (ii) (iii) (iv) (v) ∃r ∈ R≥ , ∀r ∈ R≥ , ∀r ∈ R≥ , ∀s ∈ R≥ , ∃s ∈ R≥ , ∃s ∈ R≥ , ∀s ∈ R≥ , ∃s ∈ R≥ , ∃r ∈ R≥ , ∀r ∈ R≥ , r − s = 1/2 r − s = 1/2 r − s = 1/2 r − s = 1/2 r > s ⇒ r2 − 1 > 5r. [6 marks] A3. [6 marks] A4. [6 marks] A5. [6 marks] SECTION B MATH10101 Answer THREE of the five questions B6. (i) State the induction principle for statements P (n), where n ∈ Z+ . (ii) Prove by induction on n that P 1 (a) n+1 j=1 j = 2 (n + 1)(n + 2) (b) 3 divides n3 − n for all positive integers n. [15 marks] B7. (i) State necessary and sufficient conditions on integers a, b, c, d for the fractions a/b and c/d to represent the same rational number. Describe how to express a/b in lowest terms. √ (ii) State√the equation satisfied by 3 2 which confirms that it is an algebraic number, and prove that 3 2 is not rational. (iii) Calculate, as a fraction in lowest terms, the rational number represented by the recurring infinite decimal 3.141515. [15 marks] B8. [15 marks] B9. [15 marks] B10. [15 marks] END OF EXAMINATION PAPER 3 of 3