DEFINITION:
A compound
proposition that is always true, no matters what the truth values of the
propositional variable that occur in it, is called TAUTOLOGY. A compound
proposition that is always false is CONTRADICTION neither is called CONTINGENCY.
LOGICAL
EQUIVALENCES
Compound
propositions that have the same truth values in all possible cases are called
logically equivalences .
@
The compound
propositions p and q are called logically equivalence if p↔q is a
tautology. The notation p≡q denotes that p and q are logically
equivalent.
REMARK : The sym ≡is not a logical connective and p ≡ q is not a compound proposition but rather is the
statement that p↔q is a tautology. The symbol รณ is used instead of ≡ to
denote logical equivalent.
Example:
Proof that p→q and ¬ p v q
is logically equivalent.
p
|
q
|
¬p
|
p →q
|
¬p v q
|
T
|
T
|
F
|
T
|
T
|
T
|
F
|
F
|
F
|
F
|
F
|
T
|
T
|
T
|
T
|
F
|
F
|
T
|
T
|
T
|
LOGICAL
EQUIVALENCES TABLES
LOGICAL
EQUIVALENCES TABLES INVOLVING CONDITIONAL STATEMENTS
Example: Show that
¬ ( p v ( ¬ p ^q ) ) is logically
equivalent to ¬ p ^ ¬q
using the propositional theorems and identities.
¬ ( p v ( ¬ p ^q ) ) ≡ ¬ p ^ ¬( ¬
p ^q )
≡ ¬ p ^ [¬( ¬
p) ^q ]
≡ ¬ p ^ [p ^q ]
≡ ( ¬ p ^ p)
v ( ¬ p ^ ¬q )
≡ F v ( ¬ p ^
¬q )
≡ ¬ p ^ ¬q
PROPOSTIONAL
SATISFIABILITY
- A compound proposition is satisfiable if there is an
assignment of truth values that makes it true .
- A compound proposition is unsatisfiable
if and only if its negation is true for all assignments of truth values to
the variable, that if and only if the negation is a tautology.
p/s: You can get the notes from this link ----〉1.2 Propositional Equivalences
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