Discrete
mathematics
Lecture 1
Intro to Discrete mathematics
Proposition
“
Respect all in your class
Marks Distribution
Marks
Final Term
50%
Quizes
10%
Assignments
20%
Assignments
Mid term
20%
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Quizes
Mid term
Final Term
Contact me
lailanadeem.bukc@bahria.edu.pk
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Timeline
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18th September, 23
20th November, 23
Start of semester
Course after mid
13th November, 23
15th January, 23
Midterm
Final Term
Course Outline of discrete
mathematics ( discrete structures)
01. Mathematical
reasoning
02. Basic Structures
propositional and predicate
logic, rules of inference,
proof by induction, proof by
contraposition, proof by
contradiction, proof by
implication,
Sets, Functions, Relation,
Sequences, Sums
03. Induction and
Recursion
04. Counting
05. Graphs and Trees
Mathematical Induction,
Recursive Algorithms
The Basics of Counting, The
Pigeonhole Principle,
Permutations and
Combinations
Graphs and Graph Models, Graph Terminology
and Special Types of Graphs, Representing
Graphs and Graph Isomorphism, Connectivity,
Euler and Hamilton Paths, Shortest-Path
Problems, Introduction to Trees, Applications
of Trees, Tree Traversal, Spanning Tree
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Book
Lets start the
content
Introduction
Discrete mathematics is the part of
mathematics devoted to the study
of discrete objects
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Discrete Structures/Mathematics

Discrete mathematics deals with objects that
come in discrete bundles,e.g.,1 or 2 b o o k s

Continuous mathematics deals with objects that vary continuously, e.g.,
3.42 inches from a wall.
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Why Discrete Mathematics?
• The basis of all of digital information
processing is: Discrete manipulations of discrete structures
represented in memory.
•
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It’s the basic language and conceptual foundation for all
of computer science.
Logic and Proofs
Logic is study of reasoning
e.g
A>B>C
So we can say that
A>C
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Proposition
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Examples
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Examples
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Test
• 2+ 2 = 4
• X + y =4
• Are you hungry?
Proposition
No
No
• I am Happy
Proposition
• It is raining today. proposition
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Propositional logic
 The area of logic that deals with
the proposition is called
propositional logic
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compound propositions
Many mathematical statements are
constructed by combining one or
more propositions, called
compound propositions,
formed from existing propositions
using logical operators.
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example
Simple propositions
p: “A student who has taken calculus can take this class”
q: “A student who has taken introductory computer science can
take this class.
Compound Proposition
“Students who have taken calculus or introductory computer
science can take this class”
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Boolean operators
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negation
“Michael’s PC runs Linux”
“Vandana’s
 “It is not the case that Michael’s
PC runs Linux.” This negation can
be more simply expressed as
“Michael’s PC does not run Linux.”
 ????
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memory”
smartphone has at least 32 GB of
Truth table for negation
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conjunction
Let p and q be propositions.
The conjunction of p and q,
denoted by p ∧ q, is the proposition
“p and q.”
The conjunction p ∧ q is true when
both p and q are true and is false
otherwise.
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• p: necessary to pass discrete
mathematics
• q: necessary to pass English
• p ∧ q:
• Truth table
disjunction
Let p and q be propositions.
The disjunction of p and q, denoted
by p ∨ q, is the proposition “p or q.”
The disjunction p ∨ q is false when
both p and q are false and is true
otherwise.
It is “inclusive or” means true when
both true
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• p: necessary to pass discrete
mathematics
• q: necessary to pass English
• p ∨ q:
• Truth table
Natural language is
ambiguous
• p: Leena is a singer
• q: Leena is a writer
• p or q:
Note that English “or” can be
ambiguous regarding the “both”
case
Need context to disambiguate the
meaning
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• p: Leena is a man
• q: Leena is a woman
• p or q:
Exclusive or operator
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example
• p: “A student can have a salad with
dinner”
• q: “A student can have soup with
dinner,”
•pVq
•p⊕q
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The implication
operator
• p: I am elected
• q: I will lower taxes
The conditional statement p → q is
the proposition
• If I am elected, then I will lower taxes
“if p, then q.”
The conditional statement p → q is
false when p is true and q is false,
and true otherwise.
p is called the hypothesis (premise)
q is called the conclusion (or
consequence)
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p
q
P -> q
example
“If
it is sunny, then we will go to the
beach”
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“If Maria learns discrete mathematics, then she
will find a good job.”
There are many other ways to express this
conditional statement in English. Among the
most natural of these are
• “Maria will find a good job when she learns
discrete mathematics.”
• “For Maria to get a good job, it is sufficient
for her to learn discrete mathematics.” and
• “Maria will find a good job unless she does
not learn discrete mathematics.”
CONVERSE,
CONTRAPOSITIVE,
AND INVERSE
If it is raining, then the home team wins
If p then q…….. p  q
Converse
If q then p
The proposition q → p is called the
converse of p → q.
The contrapositive of p → q is the
proposition ¬q → ¬p.
Contrapositive
-q  -p
The proposition ¬p → ¬q is called the
inverse of p → q
Inverse
-p  -q
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• “You can take the flight if and only if
you buy a ticket.”
BICONDITIONALS
Let p and q be propositions.
The biconditional statement p ↔ q is
the proposition “p if and only if q.”
The biconditional statement p ↔ q is
true when p and q have the same truth
values, and is false otherwise.
Biconditional statements are also
called bi-implications
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Truth Tables of
Compound Propositions
• (p ∨ ¬q) → (p ∧ q).
p
Construct the truth table of the
compound proposition
(p ∨ ¬q) → (p ∧ q).
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q
-q
p ∨ ¬q
p∧
q
(p ∨ ¬q) →
(p ∧ q).
Precedence of Logical
Operators
Negation
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Conjunction
Disjunction
Implication
Biconditional
Logic and Bit
Operations
Computers represent information
using bits.
A bit is a symbol with two possible
values, namely, 0 (zero) and 1
(one).
This meaning of the word bit comes
from binary digit, because zeros
and ones are the digits used in
binary representations of numbers.
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• Table for bit operator
Truth value
bits
T
F
1
0
x
y
1
1
1
0
0
1
0
0
x∨y
x∧y
X⊕y
example
Find the bitwise OR, bitwise AND,
and bitwise XOR of the bit strings
01 1011 0110 and 11 0001 1101.
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Exercise 1.1
Q.1 to 40
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