Math 132 Spring 2016
Homework I
Due: February 26, 72016 before 15:35.
i: S OA/VS
Department
o The homework consists of 4 questions.
0 Please read the questions carefully.
0 Show all your work in legibly written, wellorganized mathematical
sentences.
Math 132 Summer 2016
Midterm I
June 17, 2016
10:40 - 12:30
Name
ID#
Department
Section
0 The exam consists of 4 questions.
0 Please read the questions carefully.
0 Show all your work in legibly written, well-organized mathematical
sentences.
0 Calculato
MATH 132, Discrete and Combinatorial Mathematics, Spring 2014
Course specification
Laurence Barker, Bilkent University, version: 20 March 2014.
Course Aims: To supply an introduction to some concepts and techniques associated with
discrete mathematical me
Archive of documentation for
MATH 132, Discrete and Combinatorial Mathematics,
Bilkent University, Fall 2099, Laurence Barker
version: 11 June 2014
Source file: arch132spr14.tex
page 2: Course Specification (handout including assessment and syllabus speci
Math 132005
Quiz 4 (29/03/16)
1. Find the number of integral solutions to x1 + x2 + x3 + x4 = 19 satisfying
5 xi 10 for all i = 1, 2, 3, 4.
Letting xi = xi + 5, we obtain x1 + x2 + x3 + x4 = 39 and 0 xi 15.
Introducing the condition ci = cfw_xi 16, i = 1,
Ivlath 13202
Quiz 6
Name:
ID: \< E V
1, Let F be a simple connected planar graph with 16 edges. If F is 4-regula'r (Len each vertex has 4 edges
conning out of it)7 how many faces are there in a planar embedding of F?
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rel/CM
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Math 1 3202
Quiz 2
Name:
1. Let p, q, and 7' he primitive statements (so each can be unambiguously assigned a truth value 0 or 1).
Are the two statements
pH(q<>'r) and (qu)H
equivalent? If so7 show it; if not, give a counterexample.
P<3cfw_I%C>v> M
(3 6
Math 132005
Quiz 3 (15/03/16)
1. Let k Z+ . Prove that there exists a positive integer n such that k | n
and the only digits in n are 0s and 3s.
As it is often the case, it is much easier to prove a stronger statement: we
will find n of the form 33 . . .
hiath 132432
Quiz 4
Name: / a
IDS k E l
1. Consider the following functions:
2 loggn ~ loggloggn logg'n
n4 4' 712 21ng n log 10 n
Order these by increasing rate of growth.
W'rite f < y if f 6 0(9) but 9 Q 0(f) and f ~ 9 iff E O(g) and g E 00")
. (0 a._.
Nlath 132-02
Quiz 3
Name:
lD: i4$l
Let f : X -> Y and g : Y > Z be functions.
1. Show that if f and g are injective, then so is g o f.
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:7 39:! k5 lmiukve
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2. Show that if f and g a
Date: February 26, 2016, Friday NAME: .
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STUDENT No."ugwlugufuufzj cfw_LIA
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DEPARTMENT: .
SECTION:.O3 . _ .
_ Math 132 Spring 2016 QUIZ # 2
1) If you toss a fair coin 9 times, a) What is the probability of getting 4 tails, b) What is
the pro
lVlath 1 32- ()2
Quiz 5
m
1. Let [9n denote the nth Catalan number. Recall that these are dened by the recurrence relation [)0 : 1
TI
and bn+1 : 2 b; , anZx Let Em) denote the generating function for the sequence (39)
i:0
Translate the recurrence relation
Date: February 25, 2616, Thursday
DEPARTMENT: .
SECTION:.01 . 7
Math 132 Spring 2016 QUIZ # 2
1) If you toss a fair coin 8 times, a) What is the probability of getting 4 tails, b) What is
the probability of getting at least 4 tas, c) What is the probabi
STUDENT NO
DEPARTMENT: . ' .
Date: June 15, 2016, Wednesday NAME: . M v
1
. . ‘ ' 1
Math 132 Summer 2016 — QUIZ # 2
1) If you toss a fair coin 9 times a) what is the probability of getting 4 tails.
= fE/ﬂxfT'fowaC/‘If gar/“‘ETJw 411.42,» ear/Germ JJ
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Date: June 7, 2016, Tuesday
DEPARTMENT: .
Math 132 Summer 2016 — QUIZ # 1
1) a) Find the number of all non—negative integer solutions to the equation
‘ $1+$2+$3+$4+£E5=17.
21.204343 _
C(l7+5‘lj I7): C(Zl,’?)=W- 53575
b) Find the number of all positive i
' Date: June 24, 2016, Friday NAME .
DEPARTMENT .
Math 132 Summer 2016‘— QUIZ # 3
1) Let’
A={3024:1:+4260y| xEZ and yEZ}
and
B ='{a E A|500 _<_ a S 1400}
Calculate |.B| In other words ﬁnd the number of elements in B.
4260: I' 302444236 65 Euchre/€44 14479
Math 132 Spring 2016
Homework 111
Due: April 6, 2016 before 13:35.
Name
ID#
Section
Department
o The homework consists of 3 questions.
0 Please read the questions carefully.
0 Show all your work in legibly written, well-organized mathematical
sentence
Math 132-02: Discrete Mathematics
Problem session 3
February 25, 2016
1. The valuable hands in poker are: Pair (exactly two cars have the same value), Two Pair (two cards
have one value, two different cards have a second value, and the fifth card a third)
Math 132
Exam 2 options
1. Let Fn denote the nth Fibonacci number, defined recursively by F0 = 0, F1 = 1, and for Fn+1 =
Fn + Fn1 for n 1. Show that GCD(Fn , Fn+1 ) = 1 for all n 1.
2. (a) Let a and b be relatively prime natural numbers and c a natural nu
NAME: . STUDENT NO: .
DEPARTMENT: .
NIath 101 Section 01 ._ QUIZ # 4
SrMPl E 39mm
d .
Problem 1. Find 0?: at (:13, y) = (0, 0), if y is a diiferentiable function of :1: satisfying the equation
m+2y
/ cosa: cos(t2) dt : 3a; y.
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t I vi "3' . i. ' ;- f ," "2 f . 5; _ E J cfw_17 F'
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Math 132 Spring 2017
Homework 11
Due: April 20, 2017 before 17:00 My
A for"
Name : _ L :3 q
me : 1 '- a
Department
e The
MATH 132 DISCRETE AND COMBINATORIAL MATHEMATICS
Semester: Summer 2016
Online information:
For up to date information about the course you can use
the MOODLE system through STARS.
Exams & grading:
1st Midterm (27%) June 17, 2016, Friday, 10:4012:30
2nd Mid
BILKENT UNIVERSITY
Mathematics Department
Math 132 Discrete and Combimatorial Mathematics
Spring Semester 2013
Final Exam Solutions
1. (5 pts each) For any nonempty subset A of positive integers, we dene a graph GA with
vertex set A and the set of edges
1. (a) Prove that any subset of size 8 from the set S = {1,2,3,.,12} must contain two
elements Whose sum is 14.
I3 be a sqbset of sfze 8 from +/763 56% 5.
and H : f g f2,/2§/ film/$4,102. EEJY, MW, 777/? 7
' . [16/3]. [(6/4
- The! is a function but t -
6
1n how many ways can one move in the xyplane from (1, 2) to (9, 6) if you are allowed
to move either 1 unit up or 1 unit to the right at a time?
(b) In part (a), how many of these paths do not go through the path from (3, 3) to (4, 3) to
(5,3) to (5,4).
1. (6 pt each) For any nonempty subset A of positive integers, we define a graph GA with vertex
set A and the set of edges E given by
E = cfw_a, b A : a 6= b p prime, p - a p - b.
(a) Draw Gcfw_12,18,5,30 and determine whether it has an Euler circuit.
(b)
Math 132 Final Exam Solutions
Definition: Let A be a nonempty finite set of positive integers. Then GA will denote the undirected
graph (V, E) where
V =A
and
E = cfw_i, j | (i 6= j) (gcd(i, j) > 1) .
Q1. a) (4pts) Assuming A = cfw_2, 3, 4, 5, 6, 7, 8, 9,
Math 132-02
Quiz 3
Name:
ID:
This quiz deals with the annihilator of an element a Z/n. By definition, this is the set
AnnZ/n (a) : cfw_x Z/n|x a = 0 in Z/n.
For example, AnnZ/6 (3) = cfw_0, 2, 4 and AnnZ/7 (3) = cfw_0.
1. What is AnnZ/63 (15)? It may be h
Math 132 Final Exam
May 11, 2016
15:30 17:30
Name
ID#
Department
Section
o The exam consists of 4 questions.
0 Please read the questions carefully.
0 Show all your work in legibly written, wellorganized mathematical sentences.
0 Calculators and dict
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Name:
Department:
QUIZ 5
Show that among any n+1 integers one can nd 2 numbers so that their difference
is divisible by 71 (Use Pigeon hole principle. Identify the pigeons and pigeon holes
explicitly in your so
QUIZ 6 I . . . 1
Let A be a nonempty set and x a. sulxet'B of A. Dene the relation R 011 the
set of subsets of A as follows. .
3409,12) ganzynB,
where X and Y are subsets of A.
a) Show that R is an equivalence relation.
b) If A = cfw_1,2, 3 and B : cfw_1,
Date: Aprii 6, 2017, Thursday NAME: .
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STUDENT NO. my
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DEPARTMENT: . W's; . '. .
SECTION: . ' .
NIath 132 Spring 2017 - QUIZ # 3
1] Prove that for all n E Z*', n > 3 i 2? < n!
b 6- (#5?ng GR
n :j 3:1
Inn/31m 11*
Date: April 10, 2017, Monday NAME: .
. UT/WE
DEPARTMENT: .
SECTION: .
NIath 132 Spring 2017 QUIZ # 4
Problem: Let
A = cfw_1,233,4,5,6,7,8,9, 10
and .
R:cfw_(a,b) ] (aeA) /\ (be/l) /\ (alb) .
a) Is R a. function from A to A? (Explain)
NO! 13cquse (2,106? G
QUIZ 4
Establish the following identity by Mathematical Induction.
Ema!) : (n + 1).l 1.
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