Answer to Question #168472 in Discrete Mathematics for Muhammad Ahmad

Question

Proof the following by using logical equivalences identities. Are these system specifications consistent by using Reasoning Method? a) ¬(p ∧ (p → ¬q))→¬p

b) ¬(q →¬p)→¬q

Answer 1

(a)

$\neg(p\wedge(p\to \neg q)) \to \neg p\\ \neg(p\wedge(\neg p\vee \neg q)) \to \neg p \text{ Implication}\\ \neg((p\wedge\neg p) \vee(p\wedge\neg q)) \to \neg p \text{ Distributive }\\ \neg(F\vee(p\wedge \neg q)) \to \neg p \text{ Identity }\\ \neg(p\wedge\neg q) \to \neg p \text{ Absorption }\\ \neg\neg(p\wedge\neg q) \vee \neg p \text{ Implication }\\ (p \wedge \neg q) \vee \neg p \text{ Negation }\\ (p\vee \neg p) \wedge(\neg q \vee \neg p) \text{ Distributive }\\ T\wedge(\neg q \vee \neg p) \text{ Identity}\\ (\neg q \vee \neg p) \text{ Absorption }$

(b)

$\neg (q \to \neg p) \to \neg q\\ \neg\neg(\neg q \vee \neg p) \vee \neg q \text{ Implications }\\ (\neg q \vee \neg p) \vee \neg q \text{ Double Negation }\\ (\neg p \vee \neg q) \vee \neg q \text{ Commutative }\\ \neg p \vee (\neg q \vee \neg q) \text{ Associative }\\ \neg p \vee \neg q \text{ Idempotent Law}\\ \neg q \vee \neg p \text{ Commutative }$

The are consistent by reasoning methods