if-p-and-q-are-two-afirmation-p-q-p-q-p-q-

Question Number 1689 by 123456 last updated on 31/Aug/15

if p and q are two afirmation

\sim (p \to q) = ?

\sim (p \leftarrow q) = ?

\sim (p \Leftrightarrow q) = ?

Answered by 112358 last updated on 01/Sep/15

W e w i l l u s e t h e f a c t t h a t

p \Rightarrow q \equiv∽ p \lor q . T h e r e f o r e,

∽ (p \Rightarrow q) \equiv∽ (∽ p \lor q)

\equiv∽ (∽ p) \land ∽ q D e {M o r g a n}^{'} s l a w

\equiv p \land ∽ q . I n v o l u t i o n l a w

S i m i l a r y, ∽ (p \leftarrow q) \equiv∽ (q \Rightarrow p) \equiv q \land ∽ p .

B y d e f i n i t i o n,

p \Leftrightarrow q \equiv (p \Rightarrow q) \land (q \Rightarrow p) .

∴∽ (p \Leftrightarrow q) \equiv∽ [(p \Rightarrow q) \land (q \Rightarrow p)]

\equiv∽ (p \Rightarrow q) \lor ∽ (q \Rightarrow p)

\equiv (p \land ∽ q) \lor (q \land ∽ p)

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