Binomial Theorem and Pascals Triangle
Imdadullah Khan
Lahore University of Management Science
November 20, 2014
Imdad (LUMS)
Binomial Theorem and Pascals Triangle
November 20, 2014
1 / 54
Pascal Identity
The Pascal Identity
n1
n1
n
=
+
k
k
k 1
Imdad (
Power set
Power set of the empty set, P().
P() = cfw_ = cfw_ cfw_
20 = 1 element in P().
Power set of the set containing the , P(cfw_).
P(cfw_) = cfw_, cfw_
21 = 2 elements in P(cfw_).
Imdadullah Khan ()
Set Theory
September 17, 2014
19 / 47
September 17
Induction
Imdadullah Khan
Lahore University of Management Science
October 16, 2014
Imdad (LUMS)
Induction
October 16, 2014
1 / 53
Induction
Useful to prove propositions about all positive
integers. Examples
n
P
i = n(n+1)
2
i=0
Number of subsets of an n-e
CS-210 Discrete Mathematics
Problem Set 9 - Solutions
Student ID:
Name:
Fall 2015
Deadline: Tuesday, Dec 8, 2015(before class)
Please print out this document and do your work on the printed paper. If you want to fill this
in using a word processor, feel f
Cardinality
Imdadullah Khan
September 24, 2014
Imdadullah Khan ()
Cardinality
September 24, 2014
1 / 48
List representation of functions
Let X be set with its elements ordered
(x1, x2 . . . , ). Let f : X Y be a function. f
can be represented as a list as
Course Info
CS-210 Discrete Mathematics
Imdadullah Khan
Imdadullah Khan ()
CS-210 - Discrete Mathematics
September 10, 2014
1 / 63
Course Goals
Main Goals of CS-210 are to learn
basic principles of discrete mathematics and
their applications in computer s
Graphs
Imdad Ullah Khan
Lahore University of Management Science
November 27, 2015
Imdad (LUMS)
Graphs
November 27, 2015
1 / 119
Continuous Math. Graphs
y
y
y = f (x)
y = f (x)
x
x
Imdad (LUMS)
Graphs
November 27, 2015
2 / 119
Discrete Math. Graphs
a
d
c
b
Graphs and Trees
Imdadullah Khan
Lahore University of Management Science
November 25, 2014
Imdad (LUMS)
Graphs and Trees
November 25, 2014
1 / 118
Continuous Math. Graphs
y
y
y = f (x)
y = f (x)
x
x
Imdad (LUMS)
Graphs and Trees
November 25, 2014
2 / 118
Basic Number Theory
Imdadullah Khan
Lahore University of Management Science
December 10, 2014
Imdad (LUMS)
Basic Number Theory
December 10, 2014
1 / 112
December 10, 2014
2 / 112
Arithmetic Rules
Assume algebraic rules for +, ,
a(b + c) = ab + ac
ab = ba
Trees
Imdadullah Khan
Lahore University of Management Science
November 24, 2015
Imdad (LUMS)
Trees
November 24, 2015
1 / 52
Special Graphs : Complete Graph
The complete graph is the simple graph
containing every possible edge is present
Imdad (LUMS)
Trees
Counting
Imdadullah Khan
Lahore University of Management Science
November 13, 2014
Imdadullah Khan (LUMS)
Counting
November 13, 2014
1 / 72
Combinatorics
Combinatorics is the study of arrangement of objects.
The Art of Counting (enumerative combinatorics)
Planar Graphs
Imdadullah Khan
Lahore University of Management Science
November 27, 2015
Imdad (LUMS)
Planar Graphs
November 27, 2015
1 / 20
Planar Graphs
Is it possible to connect the three houses to the
three utilities, such that no connections cross?
Im
CS-210 Discrete Mathematics
Student ID:
Fall 2015
Name:
Problem Set 5-Solutions Deadline: Thu, Oct 22, 2015 (before class)
Please print out this document and do your work on the printed paper. If you want to fill this
in using a word processor, feel free
CS-210 Discrete Mathematics
Fall 2015
Student ID:
Name:
Problem Set 6 - Solutions Deadline: Thu, Oct 29, 2015 (before class)
Please print out this document and do your work on the printed paper. If you want to fill this
in using a word processor, feel fre
Practice Problems
December 25, 2014
Q 1. [Proofs]
Prove the following identity
n3 =
4 n+1
+n
2
3
Q 2. [Proofs]
Consider the Fibonacci series mod m. Prove that for any m, this series will
start to repeat after some time. (HINT : Think of the pigeonhole p
CS210 Discrete Mathematics
Quiz 1 Solution
University ID:
Name:
1. Show using truth table the following logical equivalence.
(P R) (Q R) (P Q) R
P
T
T
T
T
F
F
F
F
Q R
T T
T F
F T
F F
T T
T F
F T
F F
(P R) (Q R)
T
T
F
F
T
T
F
T
T
T
T
F
T
T
T
F
(P R) (Q R)
CS-210 Discrete Mathematics
Student ID:
Fall 2015
Name:
Problem Set 3 - Solutions Deadline: Tue, Oct 06, 2015 (before class)
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in using a word processor, feel fre
CS-210 Discrete Mathematics
Fall 2015
Student ID:
Name:
Problem Set 2 - Solutions
Deadline: Tuesday, Sep 29, 2015 (before class)
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in using a word processor, feel
CS-210 Discrete Mathematics
Student ID:
Fall 2015
Name:
Problem Set 4 - Solution Deadline: Tuesday, Oct 20, 2015 (before class)
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a word processor, feel
CS-210 Discrete Mathematics
Student ID:
Problem set 7 - Solutions
Fall 2015
Name:
Deadline: Thursday, Nov 12, 2015 (before class)
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in using a word processor, fee
CS-210 Discrete Mathematics
Student ID:
Problem set 8 - Solutions
Fall 2015
Name:
Deadline: Thursday, Nov 12, 2015 (before class)
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in using a word processor, fee
Logical Equivalence
Imdadullah Khan
September 10, 2014
Imdadullah Khan ()
Logical Equivalence
September 10, 2014
1 / 18
Tautology, Contradiction, Contingency
Tautology
Compound proposition whose truth value is T for all
truth values of its atomic proposit
CS-210 Discrete Mathematics
Fall 2015
Problem Set 1: Solutions
Deadline: Friday, Sep 17, 2015 (before class)
Please print out this document and do your work on the printed paper. If you want to fill this
in using a word processor, feel free to convert thi