1. Chapter 1 Propositional Logic

2. CHAPTER OUTLINE: 1.1 INTRODUCTION 1.2 PROPOSITION
1.3 COMPOUND STATEMENT
1.4 FORMAL PROPOSITION
1.5 CONDITIONAL STATEMENT
1.6 PROPOSITIONAL EQUIVALENCES

3. Propositional Logic 1.1 INTRODUCTION
Logic – used to distinguish between valid and invalid mathematical arguments.
Logic was developed by Aristotle
Application in computer science – design computer circuits, construction of computer program, verification of the correctness of programs.
Logic is a system based on proposition

4. 1.2 PROPOSITION
Proposition – is a declarative sentence either true or false, but not both.
Eg:
Tunku Abdul Rahman was the first prime minister of Malaysia - TRUE
1 + 1 = 2 - TRUE
What time is it? NOT PROPOSITION
Read this carefully. NOT PROPOSITION
x + 1 = 2 – NOT PROPOSITION
Letters are used to denote propositions – p, q, r, s.

5. 1.3 COMPOUND STATEMENTS
Many mathematical statements are constructed by combining one or more propositions.
Eg:
John is smart or he studies every night.
One or more propositions can be combining to form a single compound proposition using connectives (logical operator)

6. Common Logical connectives
Symbol
Name
And
Conjunction Or
Disjunction
Not
Negation

7. Truth table
Can be used to show how logical operators can be combine propositions to compound propositions.
Displays the truth value that correspond to all possible value (2n) of truth values for its component statement variables.
The truth value for proposition could be TRUE (T) or FALSE (F)

8. 1.4 Formal Proposition 1) Not (negation) : ~ /
Let p be a proposition. The negation of p is denoted by
, and read as “not p”.
-Eg:
Find the negation of the proposition “Today is Friday”.
The Truth Table for the Negation of a Proposition p T F

9. 1.4 Formal Proposition
2) And (conjunction) : Let p and q be propositions. The proposition of “p and q” - denoted , is TRUE when BOTH p and q are true and otherwise is FALSE. Eg: Student who have taken calculus and computer sciences can take this class. The Truth Table for the Conjunction of Two Propositions p q T F

10. 1.4 Formal Proposition
3) Or (disjunction) : Let p and q be prepositions. The preposition of “p or q” - denoted , is FALSE when BOTH p and q are FALSE and TRUE otherwise. Eg: Student who have taken calculus or computer sciences can take this class. The Truth Table for the Disjunction of Two Prepositions p q T F

11. EXAMPLE 1.1
Consider the following statements, and determine whether it is true or false.
Ice floats in water and = 4
China is in Europe and = 4
5 – 3 = 1 or 2 x 2 = 4

12. EXAMPLE 1.2
Let :
Express these propositions using p, q, r, s and logical connectives.
Anwar is a good lecturer and his students not hate mathematics
Either Hidayah is a good lecturer or she is not
Anwar’s students both hate and do not hate mathematics
Either Anwar is a good lecturer, or his students hate mathematics and Hidayah is not a good lecturer
Anwar is good lecturer but his students hate mathematics
Neither Anwar nor Hidayah is a good lecturer p
Anwar is a good lecturer q
Hidayah is a good lecturer r
Anwar’s student hate mathematics s
Hidayah’s student hate mathematics

13. EXAMPLE 1.3 Let p and q be the following propositions:
p = It is below freezing
q = It is snowing
Translate the following into logical notation, using p and q and logical connectives.
It is below freezing and snowing
It is below freezing but not snowing
It is not below freezing and it is not snowing
It is either snowing or below freezing (or both)

14. EXAMPLE 1.4

15. EXERCISE

16. 1.5 CONDITIONAL STATEMENTS
1) Implication
Let p and q be a proposition. The implication is the preposition that is FALSE when p is true, q is false. Otherwise is TRUE.
p = hypothesis/antecedent/premise
q = conclusion/consequence
Express: “ if p, then q”, “q when p”, “p implies q”
Eg: If you earn an A in logic then I will give you present.
The Truth Table for the Implication ( ) p q T F

17. 1.5CONDITIONAL STATEMENTS
2) Equivalence/ Biconditional
Let p and q be a preposition. The biconditional is the preposition that is TRUE when p and q have the same truth values, and FALSE otherwise.
Express: “ p if and only if q”
Eg: -You can take flight if and only if you buy a ticket.
-You can have dessert if and only if you finish your meal
The Truth Table for the Biconditional ( ) p q T F

18. Converse, inverse, Contrapositive
IMPLICATION: Eg: “ If it is raining, then the home team wins” CONVERSE : the converse of this implication is Eg: If the home team wins, then it is raining INVERSE: the inverse of this implication is Eg : If it is not raining, then the home team does not win. CONTRAPOSITIVE : the contrapositive of this implication is Eg : If the home team does not win, then it is not raining.

20. EXERCISE

21. EXERCISE

22. EXERCISE

23. 1.6 PROPOSITIONAL EQUIVALENCES
Logically Equivalent Two propositions p and q are said to be logically equivalent, or simply equivalent or equal, denoted by if they have identical truth tables. Example: Find the truth tables of p q
p^q
-(p^q) -p -q
-p v -q T F

24. 1.6 PROPOSITIONAL EQUIVALENCES
Tautology
A compound proposition that is always TRUE, no matter what the truth values of the propositions that occur in it.
Contains only “T” in the last column of their truth table.
Contradiction
A compound proposition that is always FALSE.
Contains only “F” in the last column of their truth table.

25. 1.6 PROPOSITIONAL EQUIVALENCES
Example: T F

26. 1.6 PROPOSITIONAL EQUIVALENCES
Contingency
A proposition that is neither a tautology nor a contradiction
A statement that can be either true or false

27. THE END