MA238 . Quiz 3 Name: Solution
February 5, 2016 _
See
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
MA238 QuizZ Name: ngti m
[5112112322. 2016 ‘55: 37,3.4l33 oi: Publch [
1. Determine whether the following pairs of graphs are isomorphic. If 50. state an isomorphism. If not, give a
reason.
a} 5 5 '1' Nerf isomorEhic.|/
.1 There are Man, possible reasons,
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
February 10, 2012
MA238 Quiz 5 Name: Mag; ,
@ 2. Determine whether the following graphs are planar or not. 7 W
If the graph is planar, redraw it with no overlapping edges.
If it is not planar, nd a subgraph homeomorphic to either K5 or K33. Do this
6.2
cfw_9.3
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
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
@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
(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
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.
‘i
PITT—H ’i ‘i i
H—CII—H H
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
gr
33.3355 wow: nuuhic:
\.
a. “EDIE; Us 6 :2. 8w
\. . + v 2 no no 235: 358.8.“ 2: mo m 32 E 355 2:0 Ban 2.: 2 “an? .o
\. 1% wU mo x53: 3:022: 05 mo _. 5:200 E wows—B 2.: mo :25 2:. E «SE? .3
\. . x 3. 20 m. 0 :o 2.592 85:35 2: we :58 E 335 2: mo .55 2: 2 2
v .5. .m .m .N .N 5 ES Q-“ .Tw NJ. «rm. «Hm. $4. 2: 23.0 E 333% SR
h n T: n M E > “823% mmgmmh 2: + U E . 22.35%. gamma» 2R EoEEEEU
nacho winneuﬁéon E 55:55
mm oozes—gum Sumac 2: «E: 932 . a N m m w w N. 0 mo “m .EoEoEEoomo mucus—Una name—u 2:2 8:3
.2.)
m
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 '
. ' ,
Winter Term, 2015
Name: Mqn_
Student Number:
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 238 Discrete Mathematics
Midterm Test February 12, 2015
Instructor: Dr. K. Cameron
Time Allowed: 80 minutes
Total Value: 70 marks
Number of Pages: 6 plus
Sun
T: +m 70; + 3+ 3. $2- 26 u
tatau. 9:1 2.2196 3. no: .3: Eu sex
0. 5.3.2: + Tagaz
n. :thc .uwuwa _2w_xwo 98:3 Evmut: c. Vdumx+axtx an 20:33 mu 555.32 _
a252,: 2: 53.: 2 Ev Ba 6 2.358 95
bummuotncz .om. oSmoLEw m_ ruemm.
Amy u id 3.: 3. muAmISI S. we $5
wuss QuizS Name: _5_0Lu_cn_ ID:
March 21, 2016
Do not use a calculator. Numerical answers can include terms such at 61 or C(n,k) and dont have to be
lied.
[7 mark 51.5:
l. (a) How many solutions are there to the equation x1+ X2+ x; = 24 where each varia