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  • Muhd. Faiz Fahimi B. Razali 033225
  • Hasyazid B. Osman 033147
  • Nurul Nabilah Bt. Azman 032236
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  • Siti Aminah Bt. Ahmad Sahrel 032806
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Sunday, 24 February 2013



Propositional Equivalences

Tautology
A compound proposition that is always true.

Contradiction
A compound proposition that is always false

Contingency
A compound proposition that is neither a tautology nor a contradiction.



Logical Equivalences

Compound proposition that have the same truth values in all possible cases. Means, compound propositions p and q are called logically equivalent if p ↔ q is a tautology.

Example of Tautology and Contradiction

p
q
p -p
p -p
T
T
T
F
F
F
T
F

*p -q is always true
*p -q is always false


Logical equivalences tables

Name
Equivalence
Identity Laws
p T p
p F p
Domination Laws
p T T
p F F
Idempotent Laws
p p p
p p p
Negation Laws
p ¬p T
p ¬p F
Double Negation Law
¬(¬p) ≡ p

Commutative Law
p q q p
p q q p
Absorption Law
p (p q) p
p (p q) p
De Morgan’s Laws
¬(p q) ≡¬p ¬q
¬(p q) ≡¬p ¬q
Associative Laws
(p q) r p (q r)
(p q) r p (q r)
Distributive laws
p (q r) (p q) (p r)
p (q r) (p q) (p r)

*T = Compound proposition is always true
*F = Compound proposition is always false


Logical Equivalences involving Conditional Statements

p → q ≡ ¬p q
p → q ≡ ¬q →¬p
p q ≡¬p q
p q ≡¬(p →¬q)
¬(p → q) ≡ p ¬q
(p → q) (p r) p (q r)
(p → r) (q r) (p q) r
(p → q) (p r) p (q r)
(p → q) (p r) p (q r)


  


Logical Equivalences involving Biconditional Statements
p ↔ q ≡ ¬p ↔¬q
¬(p ↔ q) ≡ p ↔¬q
p ↔ q ≡ (p q) (¬p ¬q)
p ↔ q ≡ (p → q) (q p)


Example Logical Equivalent.

Shows that are logical equivalents.

Double Negation Law
¬(¬p) ≡ p

p
-p
-(-p)
T
F
T
F
T
F

Commutative Law
p q q p

p
q
p q
q p
T
T
T
T
T
F
T
T
F
T
T
T
F
F
F
F

Associative Law
(p q) r p (q r)

p
q
r
(p q)
(p q) ∨ r
(q r)
p (q r)
T
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
F
F
T
T
T
T
T
F
F
F
T
T
T
T
F
F
T
F
T
T
T
F
F
F
F
F
F
F


Example Logical Equivalent involving Conditional Statement

Shows that are logical equivalents.
¬(p → q) ≡ p ¬q

p
q
¬(p → q)
¬q
p ¬q
T
T
F
F
F
T
F
T
T
T
F
T
F
F
F
F
F
F
T
F

(p → q) (p r) p (q r)

p
q
r
(p → q)
(p → r)
(p → q) (p r)
(q r)
p → (q r)
T
T
T
T
T
T
T
T
T
T
F
T
F
T
T
T
T
F
T
F
T
T
T
T
T
F
F
F
F
F
F
F
F
T
T
T
T
T
T
T
F
T
F
T
T
T
T
T
F
F
T
T
T
T
T
T
F
F
F
T
T
T
F
T


Example Logical Equivalent involving Biconditional Statement
p ↔ q ≡ (p → q) (q p)


p
q
p ↔ q
p → q
q → p
(p → q) (q p)
T
T
T
T
T
T
T
F
F
F
T
F
F
T
F
T
F
F
F
F
T
T
T
T

 

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