3120 Applied Discrete Mathematics Lab 8
Name:
November 3, 2012
Student Number:
Q1: Suppose V Z+ . Let us dene the divisibility graph of V to be the digraph (with loops) such that,
for a, b V , a b is an arc if and only if a|b. (Loops are permitted)
(a) Dr
MATH 3120 Fall 2014 Assignment 9 Solutions
1. Give two proofs: (a) algebraic; and (b) combinatorial; for the identity
P pn, kq P pn 1, kq ` kP pn 1, k 1q
Solution: Algebraic:
pn 1q!
pn 1q!
P pn 1, kq ` kP pn 1, k 1q
`k
pn 1 kq!
pn 1 pk 1qq!
pn 1q!
kpn 1q
UNIVERSITY OF MANITOBA
DATE: October 16, 2014
TERM TEST 1
PAGE: 1 of 7
TIME: 90 minutes
EXAMINER: Harland
EXAMINATION: Applied Discrete Mathematics
COURSE: MATH 3120
1. For the following parts, only the answer will be marked.
[2]
(a) What is the cardinali
MATH 3120 Fall 2014 Assignment 8 Solutions
1. What is the cocient of x5 y 2 in the expansion of each of the following expressions. (For
part (b), nd the coecient of x5 y 2 z e for the appropriate value of e.)
(a) p2x 3yq7
(b) px ` y ` 2zq9
(c) p1 ` x ` yq
DEPARTMENT OF MATHEMATICS
MATH 3120 Applied Discrete Mathematics
Fall 2014
A01:
Instructor:
Lecture:
Tutorial:
Oce hours:
N. Harland, 453 Machray Hall,
Tel: 204 474 9160, e-mail: harland@cc.umanitoba.ca
(Emails must be sent from a umanitoba.ca account or
MATH 3120 Fall 2014 Assignment 11 Solutions
1. Find the general solution to ppE 6q5 qun 0.
Solution: The characteristic equation is pr 6q5 0 which has solution r 6 with
multiplicity 5. Therefore the general solution will be
un A6n ` Bn6n ` Cn2 6n ` Dn3 6n
MATH 3120 Fall 2014 Assignment 10 Solutions
1. Let C5 t1, g, g 2 , g 3 , g 4 u.
(a) Give the multiplication table for C5 .
(b) Find the set S of all isomorphisms from C5 to itself. [HINT: if you know where
an isomorphism maps the generator, then you can k
UNIVERSITY OF MANITOBA
DATE: November 20, 2014
TERM TEST 2
PAGE: 1 of 6
TIME: 90 minutes
EXAMINER: Harland
EXAMINATION: Applied Discrete Mathematics
COURSE: MATH 3120
[5] 1. Use WARSHALLs algorithm to determine the path matrix for the digraph with vertex
3120 Applied Discrete Mathematics Lab 4
Name:
October 11, 2012
Student Number:
Q1: Argue in words that the propositions (A B) C and B C are not equivalent. (HINT: considering
their truth tables will help you decide where to start. The truth tables, howeve
MATH 3120 Fall 2014 Assignment 7 Solutions
1. At a party there are 12 girls and 9 boys. For a certain game, one must line up 5 of the
girls and 3 of the guys in a row, left to right, in such a way that no two boys are side by
side. In how many ways can th
MATH 3120 Fall 2014 Assignment 2
1. (a) The Fibonacci numbers are dened to be tfn u such that
f1 1, f2 1, fn fn1 ` fn2
for n 3. Use strong induction to show
? n
? n
1
1` 5
1 5
fn ?
2
2
5
Solution: Let P pnq be the statement
? n
? n
1 5
1
1` 5
fn ?
2
2
3120 Applied Discrete Mathematics Lab 8
Name:
November 16, 2015
Student Number:
1. (a) How many positive integers n 1000 are divisible by none of 3, 7 or 11?
(b) The Euler phi function, (n), returns the number of positive integers n which are relatively
p
3120 Applied Discrete Mathematics Lab 6
Name:
November 1, 2015
Student Number:
1. (a) Give the degree sequence (in order of decreasing degree) of the following graph:
G:
How many components does G have?
(b) Explain why none of the following are degree seq
3120 Applied Discrete Mathematics Lab 5 (some xes)
Name:
October 26, 2015
Student Number:
1. Suppose P (x, y) is an arbitrary propositional statement in which variables x and y are free and represent
integers. Let S be the following statement:
(x) (y) P (
MATH 3120 Fall 2014 Assignment 4 Solutions
Attempt all questions and show all your work. The assignment is due Thursday October
9th IN CLASS. No late assignments, including assignments handed in after class, will be
marked. (Not all questions will necessa
MATH 3120 Fall 2014 Assignment 3
1. Use truth tables to determine whether the following argument is valid
B_C
BA
C A
A
Solution: Let W stand for pB _ Cq ^ pB Aq ^ pC Aq and X stand for
ppB _ Cq ^ pB Aq ^ pC Aqq A
A
T
T
T
T
F
F
F
F
B
T
T
F
F
T
T
F
F
C
T
F
MATH 3120 Fall 2014 Assignment 6 Solutions
1. Use WARSHALLs algorithm to determine the path matrix for the diagraph with vertex
set v ta, b, c, d, e, f u and arc set
E tpa, bq, pa, dq, pd, eq, pb, eq, pd, f q, pe, f q, pb, cq, pc, aqu.
Show the matrices P
MATH 3120 Fall 2014 Assignment 5 Solutions
2
5
5
3
1. Let A be the adjacency matrix of a graph, and A
2
0
0
5
4
5
5
0
0
5
5
4
5
0
0
2
5
5
2
0
0
0
0
0
0
0
1
0
0
0
0
1
0
(a) How many paths of length 3 are there from vertex 3 to vertex 1?
Solution:
The numb
3120 Applied Discrete Mathematics Lab 6
Name:
October 22, 2012
Student Number:
Q1: (a) Write out the trace of Algorithm Gcd, with inputs (m, n) = (468, 224).
(b) Write out the trace of Algorithm Euclid, with inputs (m, n) = (468, 224).
Solution:
(a)
a
468
UNIVERSITY OF MANITOBA
MIDTERM I
DATE: Oct 18, 2012
TITLE PAGE
DEPARTMENT & COURSE NO: MATH 3120
TIME: 90 minutes
EXAMINATION: Applied Discrete Mathematics
EXAMINER: Craigen
NAME: (Print in ink)
STUDENT NUMBER:
SIGNATURE: (in ink)
(I understand that cheat
3120 Applied Discrete Mathematics Lab 10
Name:
November 24, 2013
Student Number:
1. What is the coecient of x5 y 2 in the expansion of each of the following expressions? (In (a) and (c)
that isin part (b) nd the coecient of x5 y 2 z e , for suitable value
3120 Applied Discrete Mathematics Lab 9
Name:
November 17, 2013
Student Number:
1. (a) Use Algorithm 3.1 (BUILDTREE) to construct a tree for alphabetizing the words of the next
sentence: Only the labelled tree needs to be recorded in your answer.
(b) Perf
3120 Applied Discrete Mathematics Lab 7
Name:
November 3, 2013
Student Number:
2
5
5
3
1. (a) A is the adjacency matrix of a simple graph, and A =
2
0
0
i.
ii.
iii.
iv.
5
4
5
5
0
0
5
5
4
5
0
0
2
5
5
2
0
0
0
0
0
0
0
1
0
0
0
.
0
1
0
How many paths of lengt
3120 Applied Discrete Mathematics Lab 8
Name:
November 18, 2013
Student Number:
1. Use WARSHALLs algorithm, as dened in MR and in class (notes still to come), to determine the
path matrix for the digraph with vertex set V = cfw_a, b, c, d, e, f and arc/e
3120 Applied Discrete Mathematics Lab 7
Name:
October 27, 2012
Student Number:
Q1: We have studied an algorithm, Hanoi, that solves the n-disk Towers of Hanoi problem for any value
of n. In how many (single-disk) moves does it accomplish this task, for ea
3120 Applied Discrete Mathematics Lab 11
Name:
December 1, 2013
Student Number:
1. Give two proofs: (a) algebraic; and (b) combinatorial; for the identity
P (n, k) = P (n 1, k) + kP (n 1, k 1).
Solution:
(a) P (n, k) =
n!
(nk)! .
Applying this formula to