Functions
Imdad Ullah Khan
September 28, 2016
Imdad Ullah Khan ()
Functions
September 28, 2016
1 / 57
Ordered tuples
The ordered n-tuple (a1, a2, . . . , an ) is an ordered
collection of n objects
Unlike sets order matters here
Notation: (a1, a2, . . . ,
Course Info
CS-210 Discrete Mathematics
Imdad Ullah Khan
Imdadullah Khan ()
CS-210 - Discrete Mathematics
August 30, 2016
1 / 64
Course Goals
Main Goals of CS-210 are to learn
basic principles of discrete mathematics and
their applications in computer sci
Propositional Logic
Imdad Ullah Khan
August 30, 2016
Imdad Ullah Khan ()
Propositional Logic
August 30, 2016
1 / 46
Logic
Computer Science = Problem Solving
Mathematics is at the heart of problem
solving
definition, modeling, solution, analysis
Logic is c
Induction
Imdad Ullah Khan
Lahore University of Management Science
October 17, 2016
Imdad (LUMS)
Induction
October 17, 2016
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-
Set Theory
Imdad Ullah Khan
September 8, 2016
Imdad Ullah Khan ()
Set Theory
September 8, 2016
1 / 47
Sets
A set is an unordered collection of objects
A = cfw_1, 2, a, b, Fred, LUMS
elements/members: a A, A 3 LUMS
B : the set of all professors in LUMS
B :
CHEM 531
Assignment #1 (Due date: Oct. 05, 2015, 3 pm)
Q1:
Give a systematic (IUPAC) name for the following compounds.
a)
k)
H2C
H3C
b)
(10)
C
H
CH3
H
C
H2
C
H2
C
H2
C
Cl
Cl
l)
H3C
H2
C
H2
C
H2
C
H2
C
H2C
H
C
H2
C
CH2
H3C
C
H2C
c)
m)
d)
n)
O2N
o)
CH3
O
I
5.12 Spring 2003
Outline: First Half of Semester
I. Review of Lewis Bonding Theory
A. Ionic Bonding
B. Covalent Bonding
1. Multiple Bonding
2. Formal Charge
C. Short-Hand for Chemists
1. Line-Angle Formulas
2. Dashes and Wedges
3. Curved Arrow Formalism
I
Cambridge International Examinations
Cambridge International Advanced Level
9709/72
MATHEMATICS
Paper 7 Probability & Statistics 2 (S2)
May/June 2016
1 hour 15 minutes
*1157967783*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (M
Cambridge International Examinations
Cambridge International Advanced Level
9709/32
MATHEMATICS
Paper 3 Pure Mathematics 3 (P3)
May/June 2016
1 hour 45 minutes
*5402648439*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9)
READ
Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level
9709/42
MATHEMATICS
Paper 4 Mechanics 1 (M1)
May/June 2016
1 hour 15 minutes
*8269105851*
Additional Materials:
List of Formulae (MF9)
READ THESE INSTRUCTI
Cambridge International Advanced Subsidiary and Advanced Level
9701 Chemistry June 2016
Grade Thresholds
CHEMISTRY
Cambridge International Advanced Subsidiary and Advanced Level
Grade thresholds taken for Syllabus 9701 (Chemistry) in the May/June 2016 exa
Cambridge International Advanced Subsidiary and Advanced Level
9709 Mathematics June 2016
Grade Thresholds
MATHEMATICS
Cambridge International Advanced Subsidiary and Advanced Level
Grade thresholds taken for Syllabus 9709 (Mathematics) in the May/June 20
Cambridge International Examinations
Cambridge International Advanced Level
MATHEMATICS
9709/32
May/June 2016
Paper 3
MARK SCHEME
Maximum Mark: 75
Published
This mark scheme is published as an aid to teachers and candidates, to indicate the requirements o