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ID Number: Section (circle): 1 2 3 45678 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 1 10:3011:20 C. Hewitt 2 12:30—1z20 E. Teske 3 9:30—10:20 S. Furino 4 10:30—11:20 E. Teske 5 11:3012:20 S. New 6 1:302:20 Y.R. Liu 7 2:30—3:20 R. Moosa 8 12:301220 J. Koeller 9 8:309:20 J. Koeller 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. 9 1: Recall that the symbols .1, A, V, —> and <——> are alternate notations for the connectives
NOT, AND, OR, =>, and 4:), respectively. (a) Determine whether P <——> (Q —> —:P) is equivalent to —r(P —> Q). (b) Express the statement “ m 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: ﬁ3A3V7_>7(—)’(’)’V33’0317+3x7=3<,s7m)y7z (0) Determine whether the statement “Va: :10 S a: x as ” 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 em = 0 and a1 = 1, and for n 2 2 let an = 5an_1  6an_2. Show that an = 3” — 2"
for all n 2 0. n(n+1) for alanO. [5] 2: (b) Show thatEn:(—1)ii2 = (—1)” [5] 3: (a) Find the term involving 11:1 in the expansion of (x2 + 21—m)8. 2 l
[5] (b) Evaluate the sum 2 (z n _+ ) for all n 2 0. (Prove that your answer is correct).
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 + r then gcd(a, b) = gcd(b, r). [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 ale and b1c and gcd(a, b) = d then abcd. This page may be used for rough work. It will not be marked. ...
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 Spring '10
 Mesta
 Division, Semantics

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