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MATH10101 Three hours The total number of marks on the paper is 75. A further 25 marks are available from coursework, making a total of 100. UNIVERSITY OF MANCHESTER SETS, NUMBERS AND FUNCTIONS ?? January 2011 ??.?? – ??.?? Answer ALL FIVE questions in Section A (30 marks in all) and THREE questions in Section B (45 marks in all). Electronic calculators may be used, provided that they cannot store text. 1 of 6 P.T.O. MATH10101 SECTION A Answer ALL FIVE questions A1. Construct truth tables for the statements: 1) P ⇒ Q 2) not (P and Q) 3) not (P or not Q) 4) (P ⇐ Q) and (Q ⇐ R). [6 marks] A2. Prove or disprove each of the following statements: 1) ∃p ∈ Q, ∀q ∈ Q, 3p = q 2) ∀q ∈ Q, ∃p ∈ Q, 3p = q 3) ∃q ∈ Q+ , ∀p ∈ Q+ , q ≤ p 4) ∀p, q ∈ Q+ , q ≤ p 5) ∀p, q ∈ Q+ , p2 + q 2 > 2pq. [6 marks] A3. 1) Explain what is meant by saying that Π is a partition of A. 2) Explain what is meant by an equivalence relation on A. 3) Let A = {1, 2, 3, 4, 5} . (a) If Π = {{1, 3, 5} , {2} , {4}} is a partition of A, write out the relation on A induced by Π. (b) If R = {(2, 3) , (5, 5) , (2, 2) , (3, 3) , (5, 4) , (1, 1) , (4, 5) , (3, 2) , (4, 4)} is an equivalence relation on A, write out the partition of A induced by R. [6 marks] 2 of 6 P.T.O. MATH10101 A4. 1) Find an inverse to 44 modulo 83. 2) Solve 44x ≡ 13 mod 83. 3) Solve 9x ≡ 5 mod 44. [6 marks] A5. 1) State the Binomial Theorem. 2) Find x satisfying 15 X 15 . x = 3 r r=0 2 r 3) Evaluate n X 3r 2n−r n r=0 5n r for any n ≥ 1. [6 marks] 3 of 6 P.T.O. MATH10101 SECTION B Answer THREE of the five questions B6. 1. Describe the induction principle for statements P (n), n ∈ Z+ . Explain how to rewrite it as a principle for determining when (A ⊆ Z+ ) ⇒ (A = Z+ ) is true. √ 2. Give an inductive definition of the real numbers ( 2)n and (2n)!, for all integers n ≥ 0. 3. Let qn be the real number 1 1 1 1 − 2 ... 1 − 2 ; 1− 2 2 3 n prove by induction on n that qn = (n + 1)/2n for all integers n ≥ 2. [15 marks] B7. 1. Given any two sets A and B, describe the sets A ∪ B, A ∩ B, A \ B, and A × B. 2. Show that the empty set ∅ satisfies ∅ ⊆ A, and explain what is meant by the statement that A and B are disjoint. 3. Prove that (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C) and that (A ∪ B) × C = (A × C) ∪ (B × C) for any set C. 4. Describe a counterexample to the statement that (A ∪ B) × (C ∪ D) = (A × C) ∪ (B × D) in R2 , for all subsets A, B, C, and D of R. 4 of 6 [15 marks] P.T.O. MATH10101 B8. 1) Given integers a and b > 0, Euclid’s Algorithm makes repeated application of the Division Theorem to obtain a series of equations a b r1 r2 = = = = bq1 + r1 , r1 q2 + r2, r2 q3 + r3, r3 q4 + r4, .. . 0 < r1 < b, 0 < r2 < r1 , 0 < r3 < r2 , 0 < r4 < r3 , Prove by induction, that for all i ≥ 0 there exist mi , ni ∈ Z such that ri = mi a + ni b. (Here, r0 is defined to equal b,) 2) Using Euclid’s Algorithm find the greatest common divisor of 10249 and 3589 and write your answer as an integer linear combination of 10249 and 3589. Show your calculations. [15 marks] B9. 1) Show that there is no integer a for which 4a4 + 2a2 + a ≡ 6 mod 7. 2) Prove that if π is a permutation on a set A of order d then π e = 1 if, and only if d|e. 3) Let permutations in S6 be given by σ = (6, 4, 5, 1) ◦ (3, 6, 2, 5) ◦ (1, 6) , and ρ = (1, 4, 2, 3) ◦ (1, 6, 4) ◦ (1, 6, 4, 5) . Calculate the orders of σ, ρ, σ ◦ ρ and ρ−1 ◦ σ. [15 marks] 5 of 6 P.T.O. MATH10101 B10. 1) Draw up the multiplication table for (Z∗12 , ×) . 2) Prove that, if gcd (a, m) = 1, then a is invertible mod m. (You may assume that the greatest common divisor of two integers a and b can be written as an integer linear combination of the a and b.) Deduce that the map ρa : Z∗m → Z∗m , [r]m 7→ [ar]m is a bijection. 3) State and prove Euler’s Theorem. 4) Find all solutions to x50 ≡ 3 mod 13. [15 marks] END OF EXAMINATION PAPER 6 of 6