This document discusses the topic of discrete structures which are composed of distinct, separable parts arranged in a definite pattern. Discrete structures are important as they provide the mathematical foundation for computer science topics like data structures and algorithms. They also help develop abstract thinking skills. Some essential topics covered in discrete structures include functions, relations, sets, logic, proof techniques, counting, graphs and trees, and recursion. Logic is crucial for mathematical reasoning and important for applications like program design and circuitry. Truth tables are used to show how logical operators like negation, conjunction, disjunction, exclusive or, implication, and biconditional can combine propositions.

1. Discrete
Structures
Abdur Rehman Usmani
03419019922

2. Why is it
o“Discrete” (≠ “discreet”!) - Composed of distinct, separable parts.
o“Structures” - objects built up from simpler objects according to a definite pattern.

3. Why it is important
o Provides mathematical foundation for computer science courses such as
o data structures, algorithms, relational database theory, automata theory and
o formal languages, compiler design, and cryptography,
o mathematics courses such as linear and abstract algebra, probability, logic and set
theory, and number theory.

4. What it does
o Describes processes that consist of a sequence of individual steps.
o Helps students to develop the ability to think abstractly.

5. Books
Discrete Mathematics and Its
Applications
by Kenneth H. Rosen.
6th edition, McGraw Hill
Publisher.
Discrete Mathematics
with Applications
by Susanna S.Epp
4th edition, McGraw Hill
Publisher.

6. ESSENTIAL TOPICS TO BE COVERED:
o Functions, relations and sets
o Basic logic
o Proof techniques
o Basics of counting
o Graphs and trees
o Recursion

7. Logic
oCrucial for mathematical reasoning
oImportant for program design
oUsed for designing electronic circuitry
o(Propositional )Logic is a system based on propositions.
oA proposition is a (declarative) statement that is either true or false (not both).
oWe say that the truth value of a proposition is either true (T) or false (F).
oCorresponds to 1 and 0 in digital circuits

8. The Statement/Proposition Game
“Elephants are bigger than mice.”
Is this a statement? yes
Is this a proposition? yes
What is the truth value
of the proposition? true

9. The Statement/Proposition Game
“520 < 111”
Is this a statement? yes
Is this a proposition? yes
What is the truth value
of the proposition? false

10. The Statement/Proposition Game
“y > 5”
Is this a statement? yes
Is this a proposition? no
Its truth value depends on the value of y, but this
value is not specified.
We call this type of statement a propositional
function or open sentence.

11. The Statement/Proposition Game
“Today is January 27 and 99 < 5.”
Is this a statement? yes
Is this a proposition? yes
What is the truth value
of the proposition? false

12. The Statement/Proposition Game
“Please do not fall asleep.”
Is this a statement? no
Is this a proposition? no
Only statements can be propositions.
It’s a request.

13. The Statement/Proposition Game
“If the moon is made of cheese,
then I will be rich.”
Is this a statement? yes
Is this a proposition? yes
What is the truth value
of the proposition? probably true

14. The Statement/Proposition Game
“x < y if and only if y > x.”
Is this a statement? yes
Is this a proposition? yes
What is the truth value
of the proposition? true
… because its truth value does
not depend on specific values
of x and y.

15. Combining Propositions
As we have seen in the previous examples, one or more propositions can
be combined to form a single compound proposition.
We formalize this by denoting propositions with letters such as p, q, r, s,
and introducing several logical operators or logical connectives.

16. Logical Operators (Connectives)
We will examine the following logical operators:
• Negation (NOT, )
• Conjunction (AND, )
• Disjunction (OR, )
• Exclusive-or (XOR,  )
• Implication (if – then,  )
• Biconditional (if and only if,  )
Truth tables can be used to show how these operators can combine
propositions to compound propositions.

17. Negation (NOT)
Unary Operator, Symbol: 
P  P
true (T) false (F)
false (F) true (T)

18. Conjunction (AND)
Binary Operator, Symbol: 
P Q P Q
T T T
T F F
F T F
F F F

19. Disjunction (OR)
Binary Operator, Symbol: 
P Q P  Q
T T T
T F T
F T T
F F F

20. Connectives
Let p=“It rained last night”,
q=“The sprinklers came on last night,”
r=“The lawn was wet this morning.”
Translate each of the following into English:
¬p = “It didn’t rain last night.”
r ∧ ¬p =“The lawn was wet this morning,
and it didn’t rain last night.”
¬ r ∨ p ∨ q =“Either the lawn wasn’t wet this morning, or
it rained last night, or the sprinklers came on last night.”

21. Connectives
Let p= “It is hot”
q=““It is sunny”
1. It is not hot but it is sunny.
2. It is neither hot nor sunny.
Solution
1. ⌐p∧q
2. ⌐p∧ ⌐q

22. Exclusive Or (XOR)
Binary Operator, Symbol: 
P Q PQ
T T F
T F T
F T T
F F F
• p = “I will earn an A in this course,”
• q = “I will drop this course,”
• p ⊕ q = “I will either earn an A in this
course, or I will drop it (but not both!)”
• True when exactly one of p and q is
true and is false otherwise.