Math 132 - Spring 2013 - Third Homework Assignment
Due date You must hand in your homework to your instructor before 12:30
on May 15, 2013.
Question 1) (30pts) For n 1, let an be the number of ways to write
n as an ordered sum of positive integers, where
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
Math 132 Spring 2016
Homework IV
Due: April 27, 2016 before 13:35.
Name
:
ID# :
Section :
Department :
The homework consists of 4 questions.
Please read the questions carefully.
Show all your work in legibly written, well-organized mathematical
sentenc
Math 132 Spring 2016
Homework II
Due: March 21, 2016 before 13:35.
Name
:
ID# :
Section :
Department :
The homework consists of 4 questions.
Please read the questions carefully.
Show all your work in legibly written, well-organized mathematical
sentenc
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 132 Fall 2014
Homework IV Solutions
1. Let A = cfw_1, 2, 3, 4 and B = cfw_5, 6, 7, 8, 9, 10.
(a) How many one-to-one functions f : A B satisfy none of the following
conditions f (1) cfw_5, 6 , f (2) = 7, f (3) cfw_7, 8, and f (4) cfw_8, 9, 10.
Soluti
Math 132 Fall 2014
Homework II
Due: November 21 by 9:40 am.
Name
:
ID# :
Department :
The homework consists of 4 questions.
Please read the questions carefully.
Show all your work in legibly written, well-organized mathematical
sentences.
Simplify as
Math 132 Fall 2014 Midterm II Solutions
1) (5 pts each) Dene a relation R on A = Z+ as follows:
cfw_
R = (x, y) A A p prime, (p2 | y p2 | x)
a) Is R a function from A to A? (Explain)
Solutions: No! Because (1, 2) and (1, 3) are in R but 2 = 3.
b) Is R re
HOMEWORK III SOLUTIONS
1) Let A be a set of size 8 and B = cfw_a, b, c, d, e. How many functions f from A to B has
|f 1 (a)| 1,|f 1 (b)| 1 and |f 1 (c)| is even.
Here the exponential generating function is
2
x2 x4
x2 x3
+
+ .
1+
+
+ .
f (x) = x +
2!
3!
2!
Math 132 Fall 2014
Homework I Solutions
1) a) Considering the moves
R : (x, y) (x + 1, y) and U : (x, y) (x, y + 1),
in how many ways can one go from (0, 0) to (25, 4).
Solution: This number is same as the number of arrangements of 25 Rs and 4 U s.
29 28
Date: November 19, 2014, Wednesday
NAME:.
STUDENT NO:.
DEPARTMENT:.
Math 132 Fall 2014 QUIZ # 4
Problem 1) Determine the number of positive integers n where 1 n 10000 and n is
not divisible by 2, 3 or 7.
Solution: Dene c1 : n is divisible by 3, c2 : n is
Date: November 26, 2014, Wednesday
NAME:.
STUDENT NO:.
DEPARTMENT:.
Math 132 Fall 2014 QUIZ # 5
Problem 1) Solve the recurrence relation
an 5an1 + 6an2 = 6n
where n 2 and a0 = 4 and a1 = 5.
Solution: We have r2 5r + 6 = 0 if and only if r = 2 or r = 3. He
Date: December 3, 2014, Wednesday
NAME:.
STUDENT NO:.
DEPARTMENT:.
Math 132 Fall 2014 QUIZ # 6
Problem 1) Dene G = (V, E) where
V = cfw_ f | f is a function from cfw_1, 2, 3 to cfw_4, 5
and
3
E=
cfw_f, g V
(f (i) g(i)2 = 1
(f = g)
.
i=1
In other words G
Date: December 10, 2014, Wednesday
NAME:.
STUDENT NO:.
DEPARTMENT:.
Math 132 Fall 2014 QUIZ # 7
Problem 1) Is K4,5 a planar graph? (Explain!)
Solution: No! Because K3,3 is a subgraph of K4,5 .
(1,1)
(2,1)
(3,1)
(4,1)
(1,2)
(2,2)
(3,2)
(4,2)
(5,2)
Problem
Date: October 8, 2014, Wednesday
DEPARTMENT: . .
Math 132 Fall 2014 QUIZ # 1
1) a) In how many ways can one distribute four (identical) white marbles among nine
distinct containers?
IZ'I/M',
C(msI/ 4)=C(;2/4)= 4,3,2 493
b) Find the number of all nonne
Date: November 3, 2014, Monday
STUDENT NO: . .
DEPARTMENT: . .
Math 132 Fall 2014 QUIZ # 2
1) Let A = cfw_1,2, 3,4,5,6, 7,8 and B = cfw_9,10,11,12,13,14,15.
a) How many functions are there from A to B?
W 9
b) How many onetoone functions are there from
MTH 447
Graph Theory
Fall 2004
Theorem 1. Eulers Theorem. For a connected multi-graph
G, G is Eulerian if and only if every vertex has even degree.
Proof: If G is Eulerian then there is an Euler circuit, P , in
G. Every time a vertex is listed, that accou
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
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
WM“
Math 132 Summer 20%
Homework I
Due: June 15, 2016 before 15:45-
i: 650: MF/O/t/S
Department
o The homework consists of5 questions.
a Please read the questions carefully.
0 Show all your work in legibly written, well-organized mathematical
sentences.
0
Math 132 - Spring 2013 - First Homework
Assignment
Due date You must hand in your homework to your instructor before 12:30
on March 1, 2013.
Question 1) (30pts) In the parliament of country A there are 451 seats
and three political parties X,Y,Z. How many
Math 132 - Spring 2013 - Homework Assignment 2
To be delivered to your instructors oce on April 8, 2013 by 12:40 PM.
1. How many of the equivalence relations on A = cfw_a, b, c, d, e, f have
(a) exactly 2 equivalence classes of size 3?
(b) one equivalenc
Math 132 - Spring 2013
Homework 2 Solutions
1. How many of the equivalence relations on A = cfw_a, b, c, d, e, f have
(a) exactly 2 equivalence classes of size 3?
This is the number of partitions of A with two disjoint subsets of 3 elements
each. Thus, w
Math 132 - Spring 2013 - Third Homework Solutions
Question 1) (30pts) For n 1, let an be the number of ways to write
n as an ordered sum of positive integers, where each summand is at least
2. (For example: a6 = 5 because 6=2+2+2, 6=4+2, 6=2+4, 6=3+3, 6=6
Math 132 - Spring 2013 - First Homework Assignment
Question 1) (30pts) In the parliament of country A there are 451 seats
and three political parties X.Y.Z. How many ways y. are there of di
viding up the seats such that no party has an absolute majority.
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)