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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