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Tautologies (5) [ ∼ ( p ⇒ q )] ⇔ [ p ∧ ( ∼ q )] (law of the negation of
implication, also used in the form:) (5') ( p ⇒ q ) ⇔ [( ∼ p ) ∨ q ] (6) ( p ⇔ q ) ⇔ [( p ⇒ q ) ∧ ( q ⇒ p )] (connection between
equivalence and implication) (7) [( p ⇒ r ) ∧ ( r ⇒ q )] ⇒ ( p ⇒ q ) (law of transitivity of
implication) (8) ( p ∨ p ) ⇔ p (the idempotent property of disjunction) (9) ( p ∧ p ) ⇔ p (the idempotent property of conjunction) (10) ( p ∨ 0) ⇔ p, ( p ∨ 1) ⇔ 1 (properties of tautology
and contradiction in disjunction) (11) ( p ∧ 0) ⇔ 0 , ( p ∧ 1) ⇔ p (properties of tautology
and contradiction in conjunction) (12) [ p ∨ ( q ∧ r )] ⇔ [( p ∨ q ) ∧ ( p ∨ r )] (the distributive
property of disjunction with regard to conjunction) (13) [ p ∧ ( q ∨ r )] ⇔ [( p ∧ q ) ∨ ( p ∧ r )] (the distributive
property of conjunction with regard to disjunction) (14) [ ∼ ( p ∧ q )] ⇔ [( ∼ p ) ∨ ( ∼ q )] (negation of
conjunction - de Morgan's Law) (15) [ ∼ ( p ∨ q )] ⇔ [( ∼ p ) ∧ ( ∼ q )] (negation of
disjunction - de Morgan's Law) (16) ( p ∨ q ) ⇔ ( q ∨ p ) (the commutative property of
disjunction) (17) ( p ∧ q ) ⇔ ( q ∧ p ) (the commutative property of
conjunction) (18) [( p ∧ q ) ∧ r ] ⇔ [ p ∧ ( q ∧ r )] (the associative property
of conjunction) (19) [( p ∨ q ) ∨ r ] ⇔ [ p ∨ ( q ∨ r )] (the associative property
of disjunction) (20) p ⇒ ( p ∨ q ) (21) ( p ∧ q ) ⇒ p (22) ∼ ( ∼ p ) ⇔ p (law of double negation) (23) ( p ⇒ q ) ⇒ [( p ∧ q ) ⇔ p ] (24) ( ∼ p ) ⇒ [ ∼ ( p ∧ q )] (25) { [( ∼ p ) ⇒ ( ∼ q )] ∧ [( ∼ p ) ⇒ q ] } ⇒ p (26) [ p ∧ ( p ⇒ q )] ⇒ q (the law of ”modus ponens”) (27) [( p ⇒ q ) ∧ ( ∼ q )] ⇒ ( ∼ p ) (the law of ”modus
tollens”) (28) [( p ∧ ∼ q ) ⇒ q ] ⇔ ( p ⇒ q ) (29) ( p ⇒ q ) ⇒ [( p ∧ q ) ⇔ p ] (30) ∼ 1 ⇔ 0 , ∼ 0 ⇔ 1 , p ∨ ( ∼ p ) ⇔ 1 , p ∧ ( ∼ p ) ⇔ 0
(properties of negation)
To prove whether the given statement is a tautology,
contradiction or neither, we can use the truth table.
For example, let's verify the tautology
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