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1. Determine whether the following pairs of graphs are isomorphic. If 50. state an isomorphism. If not, give a
reason.
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Winter Term, 2016
Name:
Student Number:
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 238 — Discrete Mathematics
Midterm Test — February 23, 2016
Instructor: Dr. K. Cameron
Time Allowed: 80 minutes
Total Value: 70 marks
Number of Pages: 6' plu
TGoral marks I8
MA238 QuizS Name: 65m [5,3 ID. 6.5555 5 5:5 or; 16
December 1, 2016
Do not use a calculator.
Numerical answers can include terms such at 6! or C(n,k) or 54 + 37 - 12, and dont have to be simplified.
[ 7] 1. Consider the relation .92 dened
MA238 Quiz 2 A Name: 5 [21 u 1H; 5 ID:
October 4, 2016
[5] 1. Determine whether the following pairs of graphs are isomorphic. If so, state an isomorphism. If not, give a
reason.
a. a b v w x F: V(G,) "V(Ga) defined below is an
3 e W isomorphism
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MA238 Quiz 4 Name: , 3Q (i-[an ID:
November 22, 2016
Do not use a calculator.
Numerical answers can include terms such at 6! or C(7,3) etc. and dont have to be simplied.
[a] l. A bagel shop has 5 types of bagels: plain, sesame, multigrain, cheese, and who
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3. A drawer contains a dozen brown socks and a dozen black
socks. all unmatched. A man takes socks out at random
in the dark.
:1) How many socks must he take out to be sure that he
has at least two socks of the same color?
b) How many socks mu
MA238 . Quiz 3 Name: Solution
February 5, 2016 _
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Problems3] DH - . , . .
,3 "I ,4, l5 . w the strongly connected components of the directed graph below. Don I Just circle
E 2 mer the vertex-sets of the strongly connected components — draw the componen
EXAMPLE 5
EXAMPLE 6
EXAMPLE 7
Trees as Models
Trees are used as models in such diverse areas as computer science, chemistry,
‘ i I geology, botany,
and psychology. We Will describe a variety of such models based on trees.
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Can you Iiiid u subjecl to
which graph lhcoi'y has
not been applied?
Kiuiiini
Kiu'i
FIGURE 6 An Acquaintanceship Graph.
EXAMPLE 1
EXAMPLE 2
Graph Models
Graphs are used in a wide variety of models. We began this section by describing how to construct
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Chapter 6: Counting
6.1
The Basics of Counting
Product Rule: If a procedure is carried out by performing tasks T1, T2, , Tm, and if
each task Ti, can be done in ni ways regardless of how the previous tasks were done,
then there are n1 n2 nm ways to carry
Chapter 8: Advanced Counting Techniques
8.1
Recurrence Relations
Defn. A recurrence relation for the sequence a0, a1, a2, is an equation that expresses an
in term of one or more of the previous terms of the sequence. A sequence is called a
solution of a r
Chapter 6: Counting
6.1
The Basics of Counting
Product Rule: If a procedure is carried out by performing tasks T1, T2, , Tm, and if
each task Ti, can be done in ni ways regardless of how the previous tasks were done,
then there are n1 n2 nm ways to carry
d) c,b,d,a.e,c
I03! I. Which 01° Hie 'FOllOWI'n .serLucnccs 1') a b c
041 Vcrh'ces tom 0 wall: in i'ven graph?
“but- is the lena'l'ln O‘F ﬁre wall:
a) a.e',b.c.b b) a,e.a.d,b.c,a I
c) e.b,a,d.b.e f g d
a b c 1.5 iii: Walk a
c)
frail" nth? “ b c d '
. ' ,
(0.3
ifl-n Exercises 1—4 use an adjacency list to represent the given In Exercises 22—24 draw the graph represented by the give“
graph adjacency matrix.
23.121
200
022
(f5: ls every zero—one square matrix that is symmetric and
has zeros on the diagonal t
@Show that a directed multigraph havmg no isolated ver-
tices has an Euler circuit if and only if the graph Is weakly
connected and the in degree and out-degree of each venex
are equal
“LE
@Show that a directed multigraph having no isolated ver
tices has
Chapter 6: Counting
6.1
The Basics of Counting
Product Rule: If a procedure is carried out by performing tasks T1, T2, , Tm, and if
each task Ti, can be done in ni ways regardless of how the previous tasks were done,
then there are n1 n2 nm ways to carry
Ch 6: Working
with Ranges
Cost Of A VBA Course
Total: 23 hours
4 days $3,220
o 5.75 hours per day, $805 per day, $140 per hour
More Details
Covers same topics as we do.
We give more time to some
areas.
Ranges
a major component of working with Excel
text l
Winter Tern, 2007
Name:
l b
$
Student Number:
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 238 - Discrete Mathematics
Midterm Test - February 27, 2007
Instructor: Dr. K. Cameron
Time Allowed: 80 minutes
Total Value: 65 marks
Number of Pages: 6