Using the logical equivalences show that ((~p q) (q r)) ~q is a contradiction Λ Λ Λ Λ ((~p q) (q r)) ~q = Λ Λ Λ Λ By associative laws ~p q q r ~q = Λ Λ Λ Λ By consumption laws ~p q r ~q = Λ Λ Λ By commutative laws ~p q ~q r = Λ Λ Λ By complement properties ~p F r = Λ Λ By domination laws F the logical equivalences show that (~p V q) V (p ~q) is a tautology Λ (~p V q) V (p ~q) = Λ V q) V p ) ((~p V q) V ~q)) = By distributive laws ((~p Λ V q V p ) (~p V q V ~q) = By associative laws (~p Λ V p V q ) (~p V q V ~q) = By commutative laws (~p Λ By complement properties V q ) (~p V T) = (T Λ By domination laws T T = T Λ Using the logical equivalences show that ~(p V ~q) V (~p ~q) Λ ≡~p ~(p V ~q) V (~p ~q) = Λ by De Morgan’s law (~p q) V (~p ~q) = Λ Λ by Distributive law ~p (q V ~q) = Λ by complement properties ~p T = Λ by identity laws ~p Λ
Using the logical equivalences show that ((~p Λ q) Λ (q Λ r)) Λ ~q is a contradiction ((~p Λ q) Λ (q Λ r)) Λ ~q = By associative laws ~p Λ q Λ q Λ r Λ ~q = By consumption laws ~p Λ q Λ r Λ ~q = By commutative laws ~p Λ q Λ ~q Λ r = By complement properties ~p Λ F Λ r = By domination laws F the logical equivalences show that (~p V q) V (p Λ ~q) is a tautology (~p V q) V (p Λ ~q) = By distributive laws ((~p V q) V p ) Λ ((~p V q) V ~q)) = By associative laws (~p V q V p ) Λ (~p V q V ~q) = By commutative laws (~p V p V q ) Λ (~p V q V ~q) = By complement properties (T V q ) Λ (~p V T) = By domination laws T Λ T = T Using the logical equivalences show that ~(p V ~q) V (~p Λ ~q) ≡~p ~(p V ~q) V (~p Λ ~q) = by De Morgan’s law (~p Λ q) V (~p Λ ~q) = by Distributive law ~p Λ (q V ~q) = by complement properties ~p Λ T = by identity laws ~p Λ
Test01PracticeProblemsWithSolutions.docx