Discrete Mathematics I
MATH/CSCI 2112 Assignment 1 Due 19 Sept 2014
(1) Write negations of each of the following:
(a) Roses are red and violets are blue.
(b) The bus is late or my watch is slow.
(c) If a number is prime then it is odd or it is 2.
(2) Give

Discrete Mathematics I MATH/CSCI 2112
Assignment 6 Due Fri 14 Nov 2014
Fall 2014
(1) In his book Liber Abaci (1202), Leonardo of Pisa wrote: A pair of rabbits is placed
in a walled enclosure to nd out how many ospring this pair will produce in the
course

Discrete Mathematics I MATH/CSCI 2112 Fall 2014
Assignment 3 Due Wed 15 Oct 2014
n
(1) (a) Let n N. Which is larger:
answer.
r=0
2n + 1
r
2n+1
or
j=n+1
2n + 1
? Prove your
j
(n + 1)
n
= (n + 1)
.
r
(r 1)
(Hint: consider the case of selecting a committe o

Discrete Mathematics I MATH/CSCI 2112
Assignment 4 Due Wed 21 Oct 2014
Fall 2014
(1) Let n Z+ with prime factorization n = pn1 pn2 n3 . . . pnk . How many positive
2 3
1
k
divisors of n (including 1 and n) are there? Prove your answer.
(2) Given: m, k N;

Discrete Mathematics I MATH/CSCI 2112 Fall 2014
Assignment 5 Due Fri 31 Oct 2014 - no extensions
(1) Let a, b Z+ . Dene L = cfw_sa + tb | s, t, Z be the set of all linear combinations of a, b. Let L+ = cfw_x L | x > 0 (i) Verify gcd(a, b) L+ (ii) Show
z

Discrete Mathematics I
MATH/CSCI 2112 Assignment 7 Due 24 Nov 2014
(1) Show F3n is even.
(2) Prove that F1 + F3 + F5 + . . . + F2n1 = F2n
(3) Find a closed form formula for:
a0 = 2, a1 = 2 and an = 2an1 + 15an2 for n 2
(4) Dene a sequence of integers Hn b

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 8
Due: Wednesday 21st March: 1:30 PM
Compulsory questions
1 (a) Consider the following algorithm for nding the nth bonacci number:
Input: natural number n
Output: nth Fibona

Discrete Mathematics I MATH/CSCI 2112
Assignment 3 Due Fri 23 Oct 2015
n
(1) (a) Let n N. Which is larger:
r=0
answer.
2n + 1
r
2n+1
or
j=n+1
Fall 2015
2n + 1
? Prove your
j
ANSWER:
We expand the terms of each summation, except that we (i.e. me and my
tap

Discrete Mathematics I MATH/CSCI 2112 Fall 2015
Solutions to Assignment 5 Due 30 Oct 2015
(1) Construct 2015 consecutive composite integers (recall, the problem done in class
only had 999 consecutive integers, not 1000).
ANSWER:
To obtain 2015 consecutive

Discrete Mathematics I MATH/CSCI 2112
Assignment 6 Due 6 Nov 2015
Fall 2015
(1) (a) Prove that for positive integers, m, n; gcd(m, n) = gcd(m n, n) ANSWER:
This is a rehash of the proof of Lemma 2.
Pf: Let gcd(m, n) = d gcd(m n, n) = c.
d | m d | n d | (m

Discrete Mathematics I MATH/CSCI 2112 Fall 2014
Assignment 2 Due Mon, 28 Sept 2014
(1) For each statement below, give its negation and then determine if the statement or
its negation is true
(a) a, b R, ab = a b
y
(b) x Z such that y Z, x Z
(c) b R such

Discrete Mathematics I MATH/CSCI 2112 Fall 2016
Assignment 5
Due: 28 Oct, in class.
(1) (a) Negate a Z a > 2 (a - b a - (b + 1).
(b) Prove a Z a > 2 (a - b a - (b + 1) Hint: see (a)
(2) (a) Prove that
if p is not divisible by 5 then p2 is not divisible b

Discrete Mathematics I MATH/CSCI 2112 Fall 2014
Solutions to Assignment 5 Due Fri 31 Oct 2014 - no extensions
(1) Let a, b Z+ . Define L = cfw_sa + tb | s, t, Z be the set of all linear combinations of a, b. Let L+ = cfw_x L | x > 0 (i) Verify gcd(a, b) L

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 4
Due in: Wednesday 7th February, 1:30 PM
Compulsory questions
1 Show that for all integers n, n2 is congruent to 0, 1, 2, or 4 modulo 7.
2 Show that 213 + 3241 is divisible

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 2
Due in: Wednesday 24th January, 1:30 PM
Compulsory questions
1 Which of the following are true when A = cfw_0, 1, 2 and B = cfw_1, 2, 3, 4?
Justify your answers.
(a) (n A)

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 3
Due in: Wednesday 31st January, 1:30 PM
Compulsory questions
1 Prove or disprove the following directly from the denitions:
(a) For any odd positive integer n, at least on

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 5
Due in: Friday 16th February, 1:30 PM
Compulsory questions
1 Use Euclids algorithm to nd the greatest common divisor of the following
pairs of numbers. Write down all the

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 10
Due: Wednesday 4th April: 1:30 PM
Compulsory questions
1 For each of the following relations, determine which of the four properties: reexivity, symmetry, antisymmetry, a

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 6
Due in: Monday 12th March, 1:30 PM
Compulsory questions
1 Show that if m > 1 and n > 1 are natural numbers such that 6|mn, then
it is possible to cover an m n chessboard w

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 1
Due in: Wednesday 17th January, 1:30 PM
Compulsory questions
1 Rewrite these sentences symbolically:
(a) Maths is fun but Dr Kenney is not a good lecturer.
(b) If I work v

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 9
Due: Wednesday 28th March: 1:30 PM
Compulsory questions
1 Let A = cfw_0, 1, 3, 5, B = cfw_x R|1 < x
0.3 x 3. Find:
5, and C = cfw_x R|x <
(a) A B
(b) B C
(c) A \ B
(d) A (

MATH 2112/CSCI 2112, Discrete Structures I
Winter 2007
Toby Kenney
Homework Sheet 7
Due in: Wednesday 14th March, 1:30 PM
Compulsory questions
1 Solve the following recurrence relations. i.e. nd an explicit formula for
an in terms of n, and prove that it

Discrete Mathematics I MATH/CSCI 2112
Assignment 7 Due Mon 23 Nov 2015
Fall 2015
(1) (a) The government of Elbonia has decided to issue currency only in 5 @ and 9 @ denominations.
Show that there is largest @ value that Elbonians cannot pay with this deno