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Unformatted text preview: Monday 04 February 2002
3:00 p.m. — 4:15 p.m. NAME: Id. N0.: Faculty of Mathematics
University of Waterloo MATH 135 Term Test #1
Winter 2002 Calculators are permitted. 00 Please Print l. (a) [2] Let a and b be integers. Deﬁne what it means for a to divide b. (b) [6] Determine whether or not each of the following statements is TRUE or FALSE. Write your answer
in the space provided. i.6[3
ii.4[8
iii.—3I9
iv.7[0
v.0[1
vi.0[0 2. [8] Determine whether or not the following statements are equivalent: (P AND Q) => (Q 4:» R) and ((NOT P) 0R(NOTQ))OR(Q e R) 3. Consider the statement Vx3y(:c2 + y < 0). Assume that the domain (universe of discourse) for a: and y is the
real numbers. (a) [3] Write this statement as an English sentence.
(b) [3] Is this statement TRUE or FALSE? Justify your answer. (c) [2] If the domain is changed to the set of positive integers, is the statement TRUE or FALSE? Justify
your answer. 4. (a) [5] Using the Euclidean Algorithm, calculate GCD(998, 779).
(b) [5] Find integers a: and y so that 9882? + 7793/ = GC'D(988, 779). 5. (a) [4] Let a, b and d be integers. State the formal deﬁnition of what it means for d = GC’D(a, b).
b) [3] Using this deﬁnition, prove that ifa > 0, then GCD(a, 0) = a.
(c) [6] Suppose a, b, q and r are integers that satisfy b = aq + r. Prove that GCD(a, b) = GCD((1, r). (
(d) [5] Let a and b be integers with a < b and so that a does not divide b. Consider the following list of
applications of the Division Algorithm: b = (1111 + 1'1 0 S 1'1 < (1
Cl = q2r1+r2 OST2<T1
T1 — (131°2 + T3 0 S 7'3 < r2
Til—2 : tburn—1 + 7'7; 0 S r" < rn_1
rnl : qn+1rn + 0
Explain why this algorithm (called the Euclidean Algorithm) always terminates and prove that
1",, = GC'D(a, b). 6. [6] Prove that
13+33+...+(2n—1)3=n2(2n2—1) for all positive integers n.
(Note: Clearly explain all of your steps. Marks for this question will be heavily weighted on your form and
style of presentation.) ...
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 Fall '08
 ANDREWCHILDS

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