ANSWERS, HINTS, and TYPOS for HOMEWORK 5
1. I am not sure how to approach problem 3(c). Can you help?
ANSWER: I suggest you do a few examples (like 7 choose 4 and 8 choose 3 which were
given in the assignment). Make a ring around the numbers involved. Wha
Discrete Mathematics (550.171) Exam II
Practice Problem Solutions
1. Let A = cfw_1, 2, 3 and let B = cfw_4, 5. Let R = (A A) (B B).
(a) Prove that R is an equivalence relation on A B.
ANSWER: Observe that A B = cfw_1, 2, 3, 4, 5. R is
reexive: (1,1), (2,
Discrete Mathematics (550.171) Exam I
Friday, October 04, 2013
General Directions: This exam is closed book, closed notes. You may NOT use a calculator. To receive credit for a problem you must SHOW ALL WORK. That is, show enough
steps so that the grader
Discrete Mathematics (550.171) Exam II
Practice Problems / Study Guide
General Information
The exam covers Sections 1517, 2022, 24, and 26 from the course text. You are responsible
for all the material covered from 09/27/13 to 11/01/13. This includes mate
Discrete Mathematics (550.171) Exam I
Friday, October 04, 2013
General Directions: This exam is closed book, closed notes. You may NOT use a calculator. To receive credit for a problem you must SHOW ALL WORK. That is, show enough
steps so that the grader
Discrete Mathematics (550.171) Exam I
Solutions to Practice Problems
1. Consider the true statement: If A then B.
(a) What is the contrapositive of this statement?
ANSWER: If not B then not A.
(b) What is the converse of this statement?
ANSWER: If B then
Discrete Mathematics (550.171) Exam I
Practice Problems / Study Guide
General Information
The exam covers Sections 212, and 14, from the course text. You are responsible for all the
material covered from 09/04/13 to 09/27/13. This includes material from l
Discrete Mathematics (550.171) Exam II
Friday, November 08, 2013
General Directions: This exam is closed book, closed notes. You may NOT use a calcu-
lator. To receive credit for a problem you must SHOW ALL WORK. That is, show enough
steps so that the gra
Discrete Mathematics (550.171) Exam II
Friday, November 08, 2013
General Directions: This exam is closed book, closed notes. You may NOT use a calcu-
lator. To receive credit for a problem you must SHOW ALL WORK. That is, show enough
steps so that the gra
Discrete Mathematics (550.171) Exam III
Practice Problem Solutions
1. Let t be a positive integer. Prove that if p, q1 , q2 , . . . , qt are prime numbers and
p|q1 q2 qt
then p = qi for some 1 i t.
ANSWER: We will use proof by weak induction on t.
Base C
Discrete Mathematics (550.171) Final Exam
Practice Problems / Study Guide
General Information
The exam is CUMULATIVE; however, the focus will be on the material covered since Exam
2. This includes Sections 3539, 4445, 4750, and 52 from the course text. Yo
Fundamentals of Discrete Mathematics
It is said that if we think of mathematics as a dramatic production there are three main
characters:
Denition
Theorem
Proof
Denitions. Mathematical objects come into existence by denitions. As such, mathematical den
Contradiction, Smallest Counterexample
Suppose we are trying to prove that a statement is true. One way to approach this is to suppose
that it is not true. If supposing that it is not true leads us to something that we absolutely know is
not true, then, s
D I s(,
pilE
14
ft-fittil4 Alt t
s tl:r/#b dlLtlTtlf;ts
toql
ts;rratrWal Uog*,a!
0 W cculraoldtlytla pnw lvtd
"ii*A ppsittvt ttnltqtr na A,
wii4ue ?Yiry fzutwimtau'
i\,(-4 postivt ttrttcqrrr t4a il wYi4ue ?cfw_tru friawiuwn
io',r$rrt
fuA
wY tu(fanf
tuVpry
DJJLWH MWWMU omrwork #2 cfw_on/77m;
P!
5.11 520,00? glad) X and? are 1%?!er Wrifo al/ 0/44] 4/271 my
/ Mm?)
f/WH' Tuppw 7M1 W6! am! am. By 4279541776", 1514M are my; and
m WW far-41mm mm a
IWMJWJ pf arc/7 by x, Matt xrd/y
Mumpiymg
lump; mg. (9077451413
Functions
Denition: A relation f is called a function if (a, b) f and (a, c) f imply b = c.
In other words, for every input the function generates a UNIQUE output.
Denition: Let f be a function. The set of all possible rst elements of the ordered pairs in
r)
tfi
CI
ffi ilLytwt u -frWWil&K
and ailL and b)t.Fntur
3,o6Jupp*-rhc,t at antt b uw wtatlwtil pflwirtrtyrs
'n/taf cfw_(tb) lc
B trt rua'f
wo un fint't inttgt
bt=
ut=c *'L L
ftttL
D\SLRtTE
rw
W
ax+'wrl
Io
(al,nbl= tbt)aY+ l'[$)by =rtb(xt rsJ)
L= LuYrbil
Proof by Induction
Example: Let n N. If a R and a = 1, a = 0 then
n
aj =
j=0
an+1 1
.
a1
Proof: First we establish the base case which is for the smallest number in the set under
consideration. Since the set under consideration is the natural numbers, our
A Binomial Identity
Proposition: Let n N. Then
n
k=0
n
k = n2n1 .
k
We will give two proofs, one combinatorial and the other algebraic. It should be noted that
alternative combinatorial and algebraic proofs do exist for this problem.
Combinatorial Proof:
1.1.501Le-108Fmgn 2.01 _ _ H
OM; Po 3.31 bile Solv Siva.
._._- Virer maa-adu'om 1 ' I -
W? hng (a _Pv.,g- a laraie. +;._m31c._a45mu_+hm\gl g. cfw_cimrre' o.
_ a . uglg mm. -+r ._ Jig 3m |
\- PQLQ *hg sawmimagg: C;%ht.nnib i1; 9
Discrete Mathematics (550.171)
Homework 5 (Due Friday, October 18, 2013)
Objectives: The student will be able to
prove identities involving binomial coecients
apply various counting techniques in a variety of settings (e.g., lists, sets, partitions,
etc
Suzy; MAM'Ul/IH'T/LS H0 QKLEEZJNWWJ
1) LLLFowml/I DMW fat/Min (on rummwdml mqu N, Mm 04>],
mm mam 7M8 wngm nu 5
002323041041 N) l9) IOEK (WW N) C) bgbo [mm M)
NJ (2343) N M) N I (We)
NIH) NU Nil-94)
N4 5M0 Ni? NWJJMMKIZTW
61) 45222 (mod N) 7mm ave nomu
(mar Mammal/m5 HDWWM #5 501mm;
\_
J'
1) lbw ,va mow PQNWWS 1 mg Sgt f; 2 3 m - I
r . I If [J W P,
MTV!fo yll/l? Mmy are! Pam AND J 5 714me WWI 100714
19) elf-WWWK I, 2 ,aWd 3 a M wkyfm Win .
5w" [i] 12.7 E
tact/[IVIWWWV K, mwww 4 am 100 M kmo) has
3.
Discrete Mathematics (550.171)
Homework 10 (Due Friday, December 06, 2013)
Objectives: The student will
nd square roots in Zn
work with graphs
General Directions: You must show all work and document any assumptions to receive
full credit. All problems a
Working With Sets
Proving two sets are equal Let A and B be sets. To show A = B we must show every
element of A is also in B AND that every element of B is also in A. The template is as
follows:
Suppose x A . . . Then x B.
Suppose x B . . . Then x A.
W
Discrete Mathematics (550.171) Exam I
Friday, September 30, 2011
General Directions: This exam is closed book, closed notes. You may NOT use a calcw
later. To receive credit for a probiem you must SHOW ALL WORK. That is, Show enough
steps so that the grad
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
\ /s
6
L e t A b e t he s e t o f
p e o p l e w h o a r e g o i n g t o B e e m u z f : 7t T p1e b1 7t u l H 1 A t o r n o
Discrete Mathematics (550.171) Exam II
Practice Problem Solutions
1. Let n be a positive integer. Use induction to prove that
n
X
n2 (n + 1)2
4
j3 =
j=1
ANSWER:
Base Case (n = 1): The statement is true as
13 =
12 (1 + 1)2
4
IHYP: The statement is true whe
Discrete Mathematics (550.171) Exam I
Friday, September 30, 2011
General Directions: This exam is closed book, closed notes. You may NOT use a ealeuw
later. To receive credit for a problem you must SHOW ALL WORK. That is, show enough
steps so that the gra
Discrete Mathematics (550.171) Exam I
Solutions to Practice Problems
1. Consider the true statement: If A then B.
(a) What is the contrapositive of this statement?
ANSWER: If not B then not A.
(b) What is the converse of this statement?
ANSWER: If B then