1. CPCS222 Discrete Structures I
Intro + Logic

2. Welcome to CPCS222 Meet your Instructor..
Course Syllabus will be sent by ..
Textbook: “Discrete Mathematics and Its Applications”, by Kenneth Rosen, 6th ed.
Important Dates:
1st Exam: Monday H , G
2nd Exam: Monday H , G
Important Notes:
Absence Policy
Class Behavior

3. Why Study Discrete Structures
Digital computers are based on discrete “atoms” (bits).
Therefore, both a computer’s
structure (circuits) and
operations (execution of algorithms)
can be described by discrete math.

4. Starting with Logic.. Crucial for mathematical reasoning
Used for designing electronic circuitry
Logic is a system based on propositions.
A proposition is a statement that is either true or false (not both).
We say that the truth value of a proposition is either true (T) or false (F).

5. “Elephants are bigger than mice.”
Propositional Logic..
“Elephants are bigger than mice.”
Is this a statement?
yes
Is this a proposition?
yes
What is the truth value
of the proposition?
true

6. Propositional Logic.. “520 < 111” Is this a statement? yes
Is this a proposition?
yes
What is the truth value
of the proposition?
false

7. This is called a propositional function or open sentence.
Propositional Logic..
“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.
This is called a propositional function or open sentence.

8. “Today is January 1 and 99 < 5.”
Propositional Logic..
“Today is January 1 and 99 < 5.”
Is this a statement?
yes
Is this a proposition?
yes
What is the truth value
of the proposition?
false

9. “Please do not fall asleep.” Only statements can be propositions
Propositional Logic..
“Please do not fall asleep.”
Is this a statement? no
It’s a request no
Is this a proposition?
Only statements can be propositions

10. “x<y if and only if y>x.”
Propositional Logic..
“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

11. Compound Propositions
One or more propositions can be combined to form a single compound proposition.
Formalized by:
Denoting propositions with p, q, r, and s
Logical Operators

12. 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.

13. Truth Tables Truth Table for ~p Truth Table for p q   

14. Truth Tables Truth Table for p q Truth Table for p q

15. Precedence of logical Operation
Operator
Precedence  1   2 3   4 5
Exercises:
(p   q)  q
(p  q)  (p  q)

16. Logical Operators Converse is the proposition q  p of p q
Contrapositive of p q is q  p
The proposition p  q is called the inverse of p q
Only the contrapositive always has the same truth value of p q

17. Statements and Operations
Statements and operators can be combined in any way to form new statements. P Q
PQ
 (PQ)
(P)(Q)
(PQ)(P)(Q) T F
(PQ) Ξ (P)(Q). (Equivalence) De Morgan’s laws

18. Tautologies and Contradictions
A tautology is a statement that is always true.
Examples:
R(R)
(PQ)(P)(Q)
If ST is a tautology, we write ST.
If ST is a tautology, we write ST.

19. Tautologies and Contradictions
A contradiction is a statement that is always
false.
Examples:
R(R)
((PQ)(P)(Q))
The negation of any tautology is a contra-
diction, and the negation of any contradiction is
a tautology.

20. Logical Equivalence
A compound preposition that have the same truth values are called Logically equivalent ( Ξ )
Examples:
Show that (p  q)Ξ p   q
(p  q)Ξ ( p  q) by p  qΞ  p  q
Ξ ( p)   q
Ξ p   q

21. Hints For some important equivalence check table 6 , 7, and 8 page 25
Table 5 in Section 1.2 shows many useful laws.
Exercises 1 and 7 in Section 1.2 may help you
get used to propositions and operators.

22. Lecture Summary Introducing Discrete Structures.
What is Propositional Logic.
What is a proposition.
What is a compound proposition.
What are logical operators.
What is a truth table.