Practice Midterm - Name (print): Signature: ID Number:...

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Unformatted text preview: Name (print): Signature: ID Number: Section (circle): 1 2 3 4 5 6 7 8 9 MATH 135, Algebra for Honours Mathematics Faculty of Mathematics, University of Waterloo Term Test 1, Fall Term 2009 Date: Monday, October 19 Time: 7:00 pm—8:50 pm Section Time Instructor 10:30—11:20 C. Hewitt 12:30-1:20 E. Teske 9:30—10:20 S. Furino 10:30—11:20 E. Teske 11:30-12:20 S. New 1:30—2:20 Y.-R. Liu 2:30-3:20 R. Moosa 12:30-1z20 J. Koeller 8:30-9:20 J. Koeller }—| {COOKICDCHv-PODM Pages: This test contains 8 pages, including this cover sheet and a page at the end for rough work. Instructions: Write your name, signature and ID number, and circle your section, at the top of this page. Answer all questions, and provide full explanations. If you need more space to show your work, then use the back of the previous page. Aids: Only faculty approved calculators are allowed. 1: Recall that the symbols n, A, V, —> and <—> are alternate notations for the connectives NOT, AND, OR, :=>, and <=>, respectively. (a) Determine whether P <—> (Q —> -‘P) is equivalent to —:(P —> Q). (b) Express the statement “ x is the greatest integer such that 22: S y ”, taking the universe of discourse to be Z, and using only symbols from the following list: fi7A7V3—+’H)(’)7V3370313+7X’:,<)S7‘r?y7z (c) Determine whether the statement “Va: :1: g a: x m ” is true when the universe of discourse is Z and whether it is true when the universe of discourse is R. [5] 2: (a) Let a0 = 0 and a1 : 1, and for n 2 2 let an = 5an_1 — 6an_2. Show that an 2 3" — 2” for all n 2 0. nn(n+1) [5] 2: (b) Show that Z(~1)i i2 = (—1) for all n 2 0. i=0 . 8 [5] 3: (a) Find the term involving m1 in the expanswn of (3:2 + . ” (2n+1 > for all n 2 O. (Prove that your answer is correct). 1 [5] (b) Evaluate the sum i=0 [2] 4: (a) Define the statement “a divides b”, for integers a and b. [3] (b) State the Division Algorithm. [5] (c) Prove Proposition 2.21 from the text, which states that for all integers a, b, q and r, if a = qb + 7" then gcd(a, b) :2 gcd(b, 7‘). [5] 5: (a) Let a = 231 and b = 182. Find integers s and t such that as+bt : d, where d : gcd(a, b). [5] (b) Prove that for all integers a, b and c, if a|c and He and gcd(a, b) = d then ablcd. This page may be used for rough work. It will not be marked. ...
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This note was uploaded on 12/18/2010 for the course ECONOMICS 120 taught by Professor Mesta during the Spring '10 term at Wilfred Laurier University .

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Practice Midterm - Name (print): Signature: ID Number:...

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