1
MATHEMATICS 271 L01 FALL 2014
ASSIGNMENT 2 SOLUTIONS
1. The Fibonacci sequence f1 ; f2 ; f3 ; : is de ned recursively as follows:
f1 = f2 = 1, and for n 3, fn = fn 1 + fn 2 .
n
P 2
(a) Prove by indu
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MATHEMATICS 271 L01 FALL 2016
ASSIGNMENT 4 SOLUTIONS
1. Let S = f1; 2; 3; 4; 5; 6; 7; 8; 9g and T = f1; 2; 3; 4; 5g. Let R be the relation on P (S)
de ned by:
For any X; Y 2 P (S),
XRY , jX \ T j =
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MATHEMATICS 271 L01 FALL 2015
ASSIGNMENT 2 SOLUTION
1. De ne the sequences a0 ; a1 ; a2 ; : and b0 ; b1 ; b2 ; : recursively as follows:
a0 = 0, and for n > 0, an = ab n c + ab 3n c + n, and
5
5
b0
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MATHEMATICS 271 L01 FALL 2014
ASSIGNMENT 4 SOLUTION
1. Let R be the relation on Z+
8 (a; b) ; (c; d) 2 Z+
Z+ de ned by
Z+ ,
(a; b) R (c; d) if and only if a + b
c + d.
(a) Is R re exive? symmetric?
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MATHEMATICS 271 L01 WINTER 2015
QUIZ 3 SOLUTIONS
1. The sequence a0 ; a1 ; a2 ; : is de ned by letting a0 = 0, a1 = 1 and an = 3an 1 2an 2
for all integers n
2. Prove by strong induction on n that a
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MATHEMATICS 271 L01 WINTER 2015
QUIZ 2 SOLUTIONS
1. Use the Euclidean Algorithm to nd gcd(181; 123) and nd integers x and y such that
gcd(181; 123) = 181x + 123y.
Solution: We have
181
123
58
7
2
=
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MATHEMATICS 271 L01 FALL 2016
QUIZ 3 SOLUTIONS
1. Let a0 ; a1 ; a2 ; :be the sequence de ned by a0 = 7; a1 = 4 and for integers n
an = an 1 + 6an 2 .
Prove by induction on n that an = 2 ( 3)n + 5 2n
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MATHEMATICS 271 L01 FALL 2016
ASSIGNMENT 3 SOLUTIONS
1. An urn contains ten white balls numbered from 1 to 10, and ten black balls numbered
from 1 to 10. A sample of 5 balls is chosen from the urn.
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MATHEMATICS 271 L02 FALL 2015
ASSIGNMENT 4 SOLUTIONS
1. Let T be the relation on Z de ned by:
For all x; y 2 Z, xT y if and only if 3 j x + 2y.
(a) Prove that T is an equivalence relation on Z.
Solu
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MATHEMATICS 271 FALL 2015
ASSIGNMENT 1
Due at 12:00 noon on Friday, October 2, 2015. Please hand in your assignment to
Mark Girard at the beginning of the lab on October 2. Assignments must be under
1
MATHEMATICS 271 L01 FALL 2010
ASSIGNMENT 2 SOLUTION
1. De ne the sequences a0 ; a1 ; a2 ; : and b0 ; b1 ; b2 ; : recursively as follows:
a0 = 0, and for n > 0, an = ab n c + ab 3n c + n, and
5
5
b0
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MATHEMATICS 271 L01 FALL 2013
QUIZ 2 SOLUTION
1. Prove by induction on n that
n
P
1
n
=
for all integers n
n+1
i=1 i (i + 1)
1.
Solution:
Basis: (n = 1)
1
P
1
1
1
1
=
= =
.
1 (1 + 1)
2
1+1
i=1 i (i
1
MATHEMATICS 271 L01 FALL 2013
Quiz 1 Solutions.
1. Write the negation (in good English) of each of the following statements. The answer
\ It is not the case that ." is not acceptable.
(a) For all re
MATHEMATICS 271 L01 FALL 2013
ASSIGNMENT 3 SOLUTIONS
1
1. An urn has ten black ball numbered from 1 to 10 and ten white ball numbered from 1 to
10. For each part, please give a brief explanation and s
1
MATHEMATICS 271 L01 FALL 2013
ASSIGNMENT 4 SOLUTIONS
Due at 11:00 am on Monday, December 2, 2013. Assignment must be understandable to the marker ( i.e., logically correct as well as legible ), and
1
MATHEMATICS 271 L01 FALL 2014
ASSIGNMENT 3 SOLUTIONS
1. Let S = f1000; 1001; 1002; :; 9999g. For each of the following questions, you must
explain how you got the answer.
(a) How many numbers in S h
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MATHEMATICS 271 L01 FALL 2013
QUIZ 3 SOLUTIONS
1. Suppose that the sequence e0 ; e1 ; e2 ; : is de ned by e0 = 12, e1 = 29 and for all
integers k 2, ek = 5ek 1 6ek 2 .
Prove by strong induction on n
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MATHEMATICS 271 L01 WINTER 2015
QUIZ 1 SOLUTIONS
1. Write the negation (in good English) of each of the following statements. The answer
\ It is not the case that ." is not acceptable.
(a) For all i
1
MATHEMATICS 271 L01 FALL 2014
Quiz 1 Solutions
1. Write the negation (in good English) of each of the following statements. The answer
\ It is not the case that ." is not acceptable.
(a) For all int
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MATHEMATICS 271 FALL 2015
Practice Problems 1
For each of the following statements, prove or disprove the statement. For the false
statement, write out its negation and prove that. Also, for the con
1
MATHEMATICS 271 L01 FALL 2014
QUIZ 2 SOLUTIONS
1. Use the Euclidean Algorithm to nd gcd(234; 123) and nd integers x and y such that
gcd(234; 123) = 234x + 123y.
Solution:
234
1
0
234 = 1 123 + 111
1
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MATHEMATICS 271 FALL 2015
Practice Problems 1 Solutions
1. 8n 2 Z; n2 + 2n is even.
Solution: This statement is false. Its negation is \There exists an integer n so that n2 +2n
is odd." For example,
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MATHEMATICS 271 L01 FALL 2014
QUIZ 4 SOLUTIONS
1. For each of the following questions, please give a brief explanation on how you get the
answer and simplify your answer to a number.
(a) How many 4
{go Lu "no/v
MATHEMATICS 271 L01 FALL 2013
QUIZ 4 Friday, November 22, 2013 Duration: 30 minutes.
ID#_
1. Consider the functions Z —> Z, where f = 2.1: + l and
sz ——> Zandg:
r—l
9(17): [’62 JforeacthZ
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MATHEMATICS 271 L01 FALL 2014
Quiz 3 Solutions.
1. The sequence a1 ; a2 ; a3 ; : is de ned by letting a1 = 3, a2 = 5 and ak = 3ak 1 2ak 2
for all integers k
3. Prove by strong induction on n that an
MATHEMATICS 271 WINTER 2015
Solutions to Practice Problems 2
1. x, y R, if x and y are irrational then x + y is irrational.
Solution. This statement is false, and we will prove this by proving that it
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MATHEMATICS 271 WINTER 2015
Practice Problems 3 Solutions
For each of the statements from1 to 10, prove or disprove the statement.
1. 8A Z; 9B Z so that 1 2 B A.
Solution: This statement is false. I
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MATHEMATICS 271 WINTER 2011
MIDTERM SOLUTIONS
[6] 1. Use the Euclidean algorithm to nd gcd (89; 36). Then use your work to write
gcd (89; 36) in the form 89a + 36b where a and b are integers.
Soluti
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MATHEMATICS 271 WINTER 2015
ASSIGNMENT 2 SOLUTIONS
1. The sequence a0 ; a1 ; a2 ; : is de ned by a0 = 1, and for all integers n > 0,
an = ab n c + ab 2n c + n.
2
3
(a) Find a1 ; a2 ; a3 ; a4 ; a5 ;
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MATHEMATICS 271 L01 FALL 2013
ASSIGNMENT 2 SOLUTION
1. De ne the sequences a0 ; a1 ; a2 ; : and b0 ; b1 ; b2 ; : recursively as follows:
a0 = 0, and for n > 0, an = ab n c + ab 3n c + n, and
5
5
b0
MATHEMATICS 271 FALL 2016
ASSIGNMENT 1 SOLUTIONS
1. For each of the following statements, determine whether the statement is true or
false. Prove the true statements, and for the false statements, wri
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MATHEMATICS 271 L02 Winter 2017
QUIZ 1 Thursday January 26, 2017
Duration 45 minutes
ID#
1. (4 points) Write the negation (in good English) of each of the following statements.
The answer It is not
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MATHEMATICS 271 L02 Winter 2017
QUIZ 2 Thursday February 9, 2017
Duration 45 minutes
ID#
1. (4 points) Use the Euclidean Algorithm to find gcd(228, 78) and find integers x and y
such that gcd(228, 7
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MATH 271 FALL 2017
TENTATIVE SCHEDULE
Weeks
Sept 11 - 15
Sept 18 - 22
Sept 25 - 29
Oct 2 - 6
Oct 9 - 13
Oct 16 - 20
Oct 23 - 27
Oct 30 - Nov 3
Nov 6 - 10
Nov 13 - 17
Nov 20- 24
Nov 27 - Dec 1
Dec 4
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MATHEMATICS 271 L01 FALL 2013
ASSIGNMENT 4 SOLUTIONS
Due at 11:00 am on Monday, December 2, 2013. Assignment must be understandable to the marker ( i.e., logically correct as well as legible ), and