math discrete

1. TAF 3023: DISCRETE
MATH
Power !

2. PROPOSITIONAL LOGIC
Proposition • possible condition of the world
about which we want to say
something.
Example • Apple is a fruit.

3. PROPOSITIONAL LOGIC
• Propositional • A proposition or statement is a
logic sentence which is either true or
false.
• Example • True - Apple is a fruit.
• False - Rice is a fruit.

4. PROPOSITIONAL LOGIC
• Propositional • Propositional variables use letters to
variables represent it, just as letters used to
represent numerical variables.
• Example • p, q, r, s, ………

5. PROPOSITIONAL LOGIC
Types of Truth Table
• Negation • Conjunction
Proposition Proposition

6. PROPOSITIONAL LOGIC
Types of Truth Table
• Disjunction • Exclusive Or of Two
Proposition Propositions

7. PROPOSITIONAL LOGIC
Types of Truth Table
• Conditional Statement and Biconditional
Statement
•

8. PROPOSITIONAL LOGIC
Types of Truth Table
• Compound Proposition
•

9. PROPOSITIONAL EQUIVALENCE
• Tautology
 a compound proposition that is always true.
• Contradiction
 a compound proposition that is always false.
• Contingency
 a compound proposition that contain neither true or
false that mean in its truth table have at least one true
and at least one false.

10. PROPOSITIONAL EQUIVALENCE
Examples:
p ~p p V ~p
• Tautology T F T
F T T
• Contradiction p ~p p ^ ~p
T F F
F T F
• Contingency p ~p p  ~p
T F T
F T F

11. LOGICAL EQUIVALENCE
• In logic, statements p and q are logically equivalent if
they have the same logical content
• (Mendelson 1979:56) two statements are equivalent if
they have the same truth value in every model
• Logical Equivalence Table

12. LOGICAL EQUIVALENCE
De Morgan’s Law
• Probably the most important logical
equivalence
 ￢(p ∧ q) ≡ ￢p ∨￢q
 ￢(p ∨ q) ≡ ￢p ∧￢q

13. PREDICATE AND QUANTIFIERS
Introduction:
• Predicate is an open statement or sentence that contains a finite
numbers of variables. Predicates become statement when specifies
values are substituted for the variables by certain allowable choices
of value.
• variable x - subject
Example: • greater than 3 – predicate
“x is greater
than 3” • predicate in the form of:
 P(x) – this is a unary predicate (has one
OR variable)
 P( x, y) – this is a binary predicate (has
• denote as P(x) two variables)
 P(x1, x2, x…….., xn) – this is an n-ary or n-
place predicate – (has n individual
variables in a predicate)

14. PREDICATE AND QUANTIFIERS
Quantifiers:
• Definition • a logical symbol which makes an assertion
about the set of values which make one or
more formulas true.
• universal quantifier: read for
“all”, “each”, “every”.
• existential quantifier: read for “some”
statement that is true or false.
• Example • universal - “Everyone likes cakes“.
“Not everyone likes cakes”.
• existential - “Someone likes cakes”.
“No one likes cakes”.

15. PREDICATE AND QUANTIFIERS
Examples Using Quantifiers:
Universal and Existential Quantifier
Statement: True: False:
∀xP(x) P(x) is true for every There is an x for
x. which P(x) is false.
∃xP(x) There is an x for P(x) is false for every
which P(x) is true. x.

16. PREDICATE AND QUANTIFIERS
Examples Using Quantifiers:
Universal and Existential Quantifier
Statement: True: False:
∀xP(x) x+1>x x<2
If P(x) = 1, the If P(x) = 1 or 0, the
quantification is quantification is true.
true. But If P(x) = 3, the
quantification is false.
∃xP(x) x>3 x=x+1
If P(x) = 4, the P(x) is false for all real
quantification is number.
true.