Mathematics

"MATHEMATICS
is one of the essential emanations of the human spirit, a thing to be valued in and for itself, like art or poetry."
Oswald Veblen, 1924

1.2 Propositional Equivalences


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 pq is a tautology. The notation pq 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 pq is a tautology. The symbol รณ is used instead of to denote logical equivalent.


Example: Proof that pq 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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