Answer to Question #168472 in Discrete Mathematics for Muhammad Ahmad

Question #168472

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


1
Expert's answer
2021-03-04T15:06:11-0500

(a)

¬(p(p¬q))¬p¬(p(¬p¬q))¬p Implication¬((p¬p)(p¬q))¬p Distributive ¬(F(p¬q))¬p Identity ¬(p¬q)¬p Absorption ¬¬(p¬q)¬p Implication (p¬q)¬p Negation (p¬p)(¬q¬p) Distributive T(¬q¬p) Identity(¬q¬p) Absorption \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)

¬(q¬p)¬q¬¬(¬q¬p)¬q Implications (¬q¬p)¬q Double Negation (¬p¬q)¬q Commutative ¬p(¬q¬q) Associative ¬p¬q Idempotent Law¬q¬p Commutative \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


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