Answer to Question #349982 in Discrete Mathematics for Reigne

Question

Simplify

1. p∧(p∧q)

2. ~ (~p∨q)

3. ~ (p⇒~q)

Answer 1

1. To simplify this statement we can use the associative law: $p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r$ and the idempotent law: $p ∧ p ≡ p.$

So we have:

$p∧(p∧q)≡(p∧p)∧q≡p∧q.$

2. To simplify this statement we can use the De Morgan's Law: $¬(p ∨ q) ≡ ¬p ∧ ¬q$ and the double negation law: $¬¬p ≡ p.$

So we have:

$\lnot (\lnot p∨q)≡\lnot \lnot p∧ \lnot q≡p∧ \lnot q.$

3. To simplify this statement we can use the implication law: $p → q ≡ \lnot p ∨ q$ and De Morgan's Law: $\lnot (p ∨ q) ≡ \lnot p ∧ \lnot q$ and the double negation law: $\lnot \lnot p ≡ p.$

So we have:

$\lnot (p⇒\lnot q)≡\lnot(\lnot p ∨ \lnot q)≡\lnot\lnot p \land \lnot\lnot q≡p \land q.$

Answer:

1. $p∧(p∧q)≡p∧q.$
2. $\lnot (\lnot p∨q)≡p∧ \lnot q.$
3. $\lnot (p⇒ \lnot q)≡\lnot(\lnot p ∨ \lnot q)≡\lnot\lnot p \land \lnot\lnot q≡p \land q.$