Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
4th January 2005
Munkres 22
Ex. 22.2. (a). The map p : X Y is continuous. Let U be a subspace of Y such that p1 (U ) X is open. Then f 1 (p1 (U ) = (pf )1 (U ) = id1 (U ) = U Y is open because f is continuous. Thus p : X Y is a quotient map. (b). The map
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Generating Combinations
[Algorithm 6.4.9]
Counting Methods
Input:
& The Pigeonhole Principle
r, n
Output:
all
all
rcombinations
of cfw_1,2,n
of cfw_1,2,n
in increasing
lexicographic
order.
Chapter 6
(6.8 Pigeonhole Principle)
October 16, 2009
16, 2009
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Discrete Mathematics
CS204
Lecture #26
November 9, 2009
1
at KAIST
Graphs
Denition A graph G
= (V, E ) consists of a set V of vertices and
a set E of edges that are associated to unordered pairs of vertices
in V .
A directed graph G consists of a set V of
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Selection Sort
Recurrence Relations
Chapter 7
(7.3 Applications to
the Analysis of Algorithms)
November 4, 2009
November 6, 2009
CS204 Discrete Math (Fall 2009)
Analysis of the algorithm
(selection
(selection sort)
jinah@cs.kaist.ac.kr
2
Binary Search
Wh
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Discrete Mathematics
CS204
Lecture #27
November 11, 2009
1
at KAIST
Edsger Wybe Dijkstra (19302002)
 born in the Netherlands, began programming computers while studying
theoretical physics at the University of
Leiden.
 one of the most forceful proponent
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Announcement
Schedule 2/2
Next week
week
(A bit) Longer lecture to make up (about 40
minutes of) the last missing lecture
minutes of) the last missing lecture
Monday: ~ 5 more mins
Wednesday: ~ 5 more mins
Next Friday: ~ 30 more mins
[Class starts at
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Euler Cycle/Path
End Note Questions
I couldnt understand about connected graph.
Does it have exactly one component?
Euler cycle is the cycle in a graph G that
includes all of the edges and all of the
vertices.
I couldnt understand the shortest path alg
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Schedule 2/2
Announcement
This week
week
(A bit) Longer lecture to make up (about 40
minutes of) the last missing lecture
minutes of) the last missing lecture
Monday: ~ 5 more mins
Wednesday: ~ 5 more mins
Friday: ~ 30 more mins
[Class starts at 11 a
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Extra points (!)
End note questions
Please review
Where is a graph or a tree used in a real world
problem (something other than given in our
textbook or in class)?
Solving recurrence relations
7.3
Tail recursion
recursion
Post some interesting examp
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Last Friday
Tail Recursion
Spanning Trees
Reviews
The tail recursion is referred to a special case of recursion
where the last operation of the function is a recursive
th
th
call. It is known that the tail recursions can be easily
transformed to iterat
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
End note questions
Last quiz problem
quiz problem
using 0s and 1s
Counting Methods
& The Pigeonhole Principle
Catalan number
number
Chapter 6
Midterm
(6.4 Algorithms for Generating
Permutations & Combinations)
October 21, 2009
CS204 Discrete Math (Fall
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Quiz 3
Counting Methods
1
& The Pigeonhole Principle
2
3
4
5
6
A
7
7 boxes numbered in order
boxes numbered in order
2 sets having balls
B
[1] How many ways to put the 5 balls in the set A
into the 7 boxes when at most 1 ball can be put into
a box?
Chap
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Math 441
Autumn 2007
Solutions to Homework 5
Exercises from Munkres:
23.1. Any separation of (X, T ) is also a separation of (X, T ); therefore, if (X, T )
is connected, so is (X, T ).
N
23.2. Let BN = n=1 An . We prove by induction on N that BN is connec
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
April 21, 2006
Munkres 29
Ex. 29.1. Closed intervals [a, b] Q in Q are not compact for they are not even sequentially compact [Thm 28.2]. It follows that all compact subsets of Q have empty interior (are nowhere dense) so Q can not be locally compact. To
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
1st December 2004
Munkres 35
Ex. 35.3. Let X be a metrizable topological space. (i) (ii): (We prove the contrapositive.) Let d be any metric on X and : X R be an unbounded realvalued function on X . Then d(x, y ) = d(x, y ) + (x) (y ) is an unbounded m
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Relations
A relation R on a set X is reflexive
if (x,x) R for every x X.
A relation R on a set X is symmetric
if for all x, y X, if (x,y) R, then (y,x) R.
A relation R on a set X is antisymmetric
if for all x, y X, if (x,y) R and (y,x) R, then x=y.
if
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Why is the mathematical induction
a valid proof technique?
valid
How many relations on A = cfw_a,b ?
How many relations are there on an nelement set? 2
n2
Suppose we know P(1) is true and that
proposition P(k)
proposition P(k) P(k+1) is true for all po
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Example 4.2.3
Algorithms
val = 16
Chapter 4
(4.3 Analysis of Algorithms)
September 30, 2009
30 2009
i=5
j=4
j=3
j=2
CS204 Discrete Math (Fall 2009)
Insertion sort
2
Analysis of an Algorithm
Lets analyze the algorithm.
The for loop always executes n1 tim
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
End Note Questions
How to find an appropriate (most close) asymptotic upper/lower bound
functions?
Why we use bigO ?
How to get if we have different and O functions?
If the complexities for the best case and the worst case are different, how do
we say ab
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Schedule 1/2
Counting Methods
& The Pigeonhole Principle
Chapter 6
(6.1 Basic Principles)
October 7, 2009
2009
Wednesday 11:00 ~ 12:50
Counting
Question
How many objects/ways/steps are there?
How many different
dinners consist of one
main course and one
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Discrete Mathematics
CS204
Lecture #16
October 9, 2009
1
at KAIST
Permutations
Denition A permutation of n distinct elements x1 , . . . , xn is an
ordering of the n elements x1 , . . . , xn .
 Giving an ordering on cfw_x1 , . . . , xn is equivalent to s
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Extra points (!)
Please either send me email to let me know who you are,
or identify yourself (your name) on your posting,
so that can give you the extra points!
so that I can give you the extra points!
Trees
Where is a graph or a tree used in a real wor
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Decisions, decisions, decisions
decisions, decisions
Trees
Chapter 9
( 9.8 Decision Tree
9.10 Game Tree )
November 30, 2009
CS204 Discrete Math (Fall 2009)
Harpers experiment
jinah@cs.kaist.ac.kr
Harpers experiment
(1982)
Do animals make rational decisi
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Extra points
End Note Comments
Where is a graph or a tree used in a real world
problem (something other than given in our
problem (something other than given in our
textbook or in class) ?
Homework is difficult
Please upload solution.
Pl
Homework is
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2013
Homework #12 (due 12/7)
[9.7]
[9.8]
[9.9] # 19, 24
#
3,
#
9,
16,
22,
18
34
#E1: Draw all the spanning trees of K2,2 and indicate the isomorphism classes of these.
How many isomorphism classes are there?
#E2: How many graphs have n vertices labeled v1, v2,
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2014
*MAS201 Differential Equations and Applications_2014 Spring_v.1*
MAS201A[LEETu,TH9:00]
MAS201B[CHOITu,TH13:00]
MAS201C[TOMOOM,W13:00]
MAS201D[SCHWEIZERM,W10:30]
1.
Course Outline: This course introduces the basics of differential equations. The to
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2014
MAS201 Differential Equations and Applications
2015 Spring Semester
Class
A
B
C
D
Tu,
Tu,
Mo,
Mo,
Lecture hours
Th 10:30AM11:45AM
Th 09:00AM10:15AM
We 13:00PM14:15PM
We 09:00AM10:15AM
Lecture room
E11410
E11?
E11?
E11?
Lecturer
Kwon, Kil Hyun
Ref.
Korea Advanced Institute of Science and Technology
MAS 223

Spring 2014
MAS201 Differential Equations and Applications
2015 Spring Semester
Class
A
B
C
D
Tu,
Tu,
Mo,
Mo,
Lecture hours
Th 10:30AM11:45AM
Th 09:00AM10:15AM
We 13:00PM14:15PM
We 09:00AM10:15AM
Lecture room
E11410
E11?
E11?
E11?
Lecturer
Kwon, Kil Hyun
Ref.