logic parti
TRANSCRIPT
LOGIC
PROPOSITIONAL LOGIC
l Proposition
l declarative statement that is either true or false, but not both
l the following statements are propositions:
l The square root of 2 is irrational.
l In the year 2010, more Filipinos will go to Canada.
l -5 < 75
l the following statements are NOT propositions:
l What did you say?
l This sentence is false.
l x = 6
PROPOSITIONAL LOGIC
l Notation
l atomic propositions
l capital letters
l compound propositions
l atomic propositions with logical connectives
l defined via truth tables
l truth values of propositions
l 1 or T (true)
l 0 or F (false)
LOGICAL CONNECTIVES
Operator Symbol Usage
Negation ¬ not
Conjunction ∧ and
Disjunction ∨ or
Conditional → if, then
Biconditional ↔ iff
NEGATION
l turns a false proposition to true and turns a true proposition to false
l truth table
P ¬ P
1 0
0 1
l examplel P: 10 is divisible by 2.
l ¬ P: 10 is not divisible by 2.
CONJUNCTION
l truth table
P Q P ∧ Q
1 1 1
1 0 0
0 1 0
0 0 0
CONJUNCTION
l examples
l 6 < 7 and 7 < 8
l 2*4 = 16 and a quart is larger than a liter.
l P: Barrack Obama is the American president.
Q: Benigno Aquino III is the Filipino president.
R: Corazon Aquino was an American president.
P ∧ Q P ∧ R R ∧ Q
DISJUNCTION
l truth table
P Q P ∨ Q
1 1 1
1 0 1
0 1 1
0 0 0
DISJUNCTION
l examples
l 6 < 7 or Venus is smaller than earth.
l 2*4 = 16 or a quart is larger than a liter.
l P: Slater Young is a millionaire.
Q: Lucio Tan is a billionaire
R: Steve Jobs was a billionaire.
P ∨ Q P ∨ R R ∨ Q
CONDITIONAL/IMPLICATION
l P is the hypothesis or premise
l Q is the conclusion
l truth table
P Q P → Q
1 1 1
1 0 0
0 1 1
0 0 1
CONDITIONAL/IMPLICATION
l other ways to express P → Q:
l If P then Q
l P only if Q
l P is sufficient for Q
l Q if P
l Q whenever P
l Q is necessary for P
CONDITIONAL/IMPLICATION
l examples:
l If triangle ABC is isosceles, then the base angles A and B are equal.
l 1+2 = 3 implies that 1 < 0.
l If the sun shines tomorrow, I will play basketball.
l If you get 100 in the final exam, then you will pass the course.
l If 0 = 1, then 3 = 9.
BICONDITIONAL
l logically equivalent to P → Q ∧ Q → P
l truth table
P Q P ↔ Q
1 1 1
1 0 0
0 1 0
0 0 1
BICONDITIONAL
l examples
l A rectangle is a square if and only if its diagonals are perpendicular.
l 5 + 6 = 6 if and only if 7 + 1 = 10.
OTHER CONCEPTS
l contrapositive
l ¬ Q → ¬ P contrapositive of P → Q
l ¬ Q → ¬ P is equivalent to P → Q
l inverse
l ¬ P → ¬ Q is the inverse of P → Q
l P → Q is not equivalent to its inverse
l converse
l Q → P is the converse of P → Q
OTHER CONCEPTS
l types of propositional forms
l tautology – a proposition that is always true under all possible combinations of truth values for all component propositions
l contradiction – a proposition that is always false under all possible combinations of truth values for all component propositions
l contingency – a proposition that is neither a tautology nor a contradiction
SAMPLE TRUTH TABLES
P Q P ∧ Q (P ∧ Q) → P
1 1 1 1
1 0 0 1
0 1 0 1
0 0 0 1
(P ∧ Q) → P
SAMPLE TRUTH TABLES
P ¬ P P ∧ ¬ P
1 0 0
0 1 0
P ∧ ¬ P
SAMPLE TRUTH TABLES
P Q P ∨ Q (P ∨ Q )→ P
1 1 1 1
1 0 1 1
0 1 1 0
0 0 0 1
(P ∨ Q) → P
SAMPLE TRUTH TABLES
P Q P ↔ Q P ∧ Q ¬ P ∧¬ Q (P ∧ Q) ∨(¬ P ∧¬ Q)
1 1 1 1 0 1
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 1 1
Show that (P ↔ Q) ↔(( P ∧ Q) ∨ (¬ P ∧¬ Q))
Equivalent Propositions (Logical Equivalence)
l When are two propositions equivalent?
Suppose P and Q are compound propostions, P and Q are equivalent if the truth value of P is always equal to the truth value of Q for all the permutation of truth values to the component propositions
Equivalent Propositions(Logical Equivalence)
l Suppose P is equivalent to Q. P may be used to replace Q or vice versa.
l The Rules of Replacement are equivalent propositions(Logically equivalent propositions)
l The Rules of Replacement are used to simplify a proposition (Deriving a proposition equivalent to a given proposition)
Rules of Replacement
1. Idempotence
P ≡ ( P ∨ P ) , P ≡ ( P ∧ P )
2. Commutativity
( P ∨ Q ) ≡ ( Q ∨ P ), ( P ∧ Q ) ≡ ( Q ∧ P )
3. Associativity,
( P ∨ Q ) ∨ R ≡ P ∨ ( Q ∨ R ),
( P ∧ Q ) ∧ R ≡ P ∧ ( Q ∧ R )
4. De Morgan’s Laws
¬ ( P ∨ Q ) ≡ ¬P ∧ ¬Q,
¬ ( P ∧ Q ) ≡ ¬P ∨ ¬Q
Rules of Replacement
5. Distributivity of ∧ over ∨
P ∧ ( Q ∨ R ) ≡ ( P ∧ Q ) ∨ ( P ∧ R )
6. Distributivity of ∨ over ∧
P ∨ ( Q ∧ R ) ≡ ( P ∨ Q ) ∧ ( P ∨ R )
7. Double Negation
¬ (¬ P) ≡ P
8. Material Implication
( P ⇒ Q ) ≡ (¬ P ∨ Q )
9. Material Equivalence
( P ⇔ Q ) ≡ ( P ⇒ Q ) ∧ ( Q⇒P )
Rules of Replacement
10. Exportation
[ ( P ∧ Q ) ⇒ R ] ≡ [ P ⇒ ( Q ⇒ R ) ]
11. Absurdity
[ ( P ⇒ Q ) ∧ ( P ⇒ ¬ Q )] ≡ ¬ P
12. Contrapositive
( P ⇒ Q ) ≡ (¬ Q ⇒ ¬P )
Rules of Replacement
13. Identities
P ∨ 1 ≡ 1 P ∧ 1 ≡ P
P ∨ 0 ≡ P P ∧ 0 ≡ 0
P ∨ ¬P ≡ 1 P ∧ ¬P ≡ 0
¬0 ≡ 1 ¬1 ≡ 0