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 induction on n that
fi = fn+1 fn for all integers n 1.
i=1
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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 understandable to the marker ( i.e., logically correct as we
1
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 that en = 5 3n + 7 2n for all integers n 0.
Solution: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 integers a; b and c, if a j (b + 2c) and a j (2b + c) th
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 integers x, x2 + 3x + 2 > 0.
Solution: There is an integer
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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.
Solution: We have
89
36
17
2
= 2 36 + 17
= 2 17 + 2
= 8 2 + 1
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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 = 2n + 1 for all integers
n 1.
Solution:
Base cases (n
{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.
Is 9 one~to—one? Prove your answer.
a; [A M m ~— 7&3
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MATHEMATICS 271 L01 FALL 2010
ASSIGNMENT 4 SOLUTIONS
1. Let f : A ! B be a function and let S be a relation on B. Let R be the relation on
A de ned by \For all x; y 2 A,
(x; y) 2 R if and only if (f (x) ; f (y) 2 S." Prove or
disprove each of the follow
1
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 digit positive integers can be formed using the digits
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MATHEMATICS 271 WINTER 2011
MIDTERM
[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.
[6] 2. You are told that X is a set, and that f1; 2g 2 P (X) but f1; 2;
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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 conditional statements,
write out the converse and the con
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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, in the case n = 1, we have n2 + 2n = 1 + 2 = 3 which i
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 its negation is
true. Its negation is
x, y R such that x
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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. Its negation is: \9A Z so that 8B Z, 1 2 B A.",
=
and a
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MATHEMATICS 271 L02 FALL 2015
ASSIGNMENT 2
Due at 12:00 noon on Friday, October 16, 2015. Please hand in your assignment
to Mark Girard at the beginning of the lab on October 16. Assignments must be understandable to the marker ( i.e., logically correct
1
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)
Inductive step: Let k 1 be an integer and suppose
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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 real numbers x and y, if x is rational and y is irrationa
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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 an = 2n 1 for all integers
n 0.
Solution:
Bases (n = 0;
MATH 271 L01
QUIZ 4 SOLUTIONS
[3] 1. Let n be a positive integer. Find a simple formula (no sums of more than two
terms) for the number of subsets of cfw_1, 2, 3, . . . , 2n which contain at least one even
integer.
Solution. There are 22n = 4n subsets of
MATH 271 L01
QUIZ 2 SOLUTIONS
1. Use the Euclidean algorithm to nd gcd(45, 31), and then use your work to write
gcd(45, 31) in the form 45a + 31b for integers a and b.
Solution 1. We get
45
31
14
3
=
=
=
=
1 31 + 14
(and so 14 = 45 31)
2 14 + 3
(and so 3
MATH 271 L01
QUIZ 5 SOLUTIONS
1. Let A = cfw_1, 2, 3 and B = cfw_1, 2, 3, 4.
[2] (a) Give an example of a one-to-one function f : A B.
Solution: One example is f dened by f (1) = 1, f (2) = 2, f (3) = 3.
[2] (b) Give an example of an onto function g : B A
MATH 271 L01
QUIZ 3 SOLUTIONS
1. The sequence a0 , a1 , a2 , . . . is dened by:
a0 = 0,
a1 = 2,
and an = 3an1 + an2 for all integers n 2.
Use strong induction and the denition of even integer to prove that an is even for all
integers n 0.
Solution. Base S
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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 = 2, b1 = 3 and for n > 1, bn = 3bn 1 2bn 2 .
(a) Find
MATH 271 L01
QUIZ 1 SOLUTIONS
1. Disprove the statement: a, b Z, if a|b then (a + 1)|(b + 1).
Solution. A counterexample is a = 1, b = 2. Then a|b (since 1|2), but (a + 1) | (b + 1)
(since 2 | 3).
You could also write out the negation of this statement (w
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MATHEMATICS 271 WINTER 2015
ASSIGNMENT 1 SOLUTIONS
1. In this question, you can do any part using the result in the previous part(s).
(a) Prove by contradiction that for all integers n, if 3 j n2 then 3 j n.
Solution: Suppose that n is an integer so tha
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MATHEMATICS 271 L01 FALL 2014
ASSIGNMENT 1 SOLUTION
1. For each true statement, give a proof. For each false statement, write the negation and
prove that.
(a) 8x, y 2 R, if bxyc = bxc byc then x 2 Z or y 2 Z.
(b) 8x 2 R, 9y 2 Z so that bxyc = bxc byc.
(
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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 = jY \ T j.
(a) Prove that R is an equivalence relation.
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, write down their negations
and prove them. Note that you a
1
MATHEMATICS 271 L01 FALL 2016
ASSIGNMENT 2 SOLUTION
1. The Fibonacci sequence f1 ; f2 ; f3
fk = fk 1 + fk 2 .
(a) Prove that fn <
(a) Basis (n = 2; 3)
f2 = 1 < 74 = 74
n 1
7
4
1
for all integers n
2 1
7
4
=
is de ned by f1 = f2 = 1 and for integers k
3,
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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.
(a) How many di erent samples are there?
Solution: Ther
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MATHEMATICS 271 WINTER 2016
Practice Problems 2 Solutions
For each of the following
statements, prove or disprove the statement. Note that you
p
can use the fact that 2 is irrational. For all other irrational numbers, you must prove
that they are in fac