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Tutorial 2 1. Convert the following expression into Conjunctive Normal Form (CNF) (P → Q) V ¬R. Answer: (¬P V Q) V ¬R ¬ (¬P V Q) ∧¬ ¬R (P ∧ ¬Q) ∧ R

Useful Double negation De morgan’s law

Cannot be solved

2. How to find the disjunctive normal form (DNF) of (p v q) → ¬ r by using truth table

p

q

r

(p V q)

¬r

(p v q) → ¬ r

T

T

T

T

F

F

T

T

F

T

T

T

T

F

T

T

F

F

T

F

F

T

T

T

F

T

T

T

F

F

F

T

F

T

T

T

F

F

T

F

F

T

F

F

F

F

T

T

(p v q) → ¬ r = (p ∧ q ∧ ¬r) V (p ∧ ¬q ∧ ¬r) V (¬p ∧ q ∧ ¬r) V (¬p ∧ ¬q ∧ r) V (¬p ∧ ¬q ∧ ¬r)

3. Show that ¬(p v (¬ p ∧ q)) and ¬p ∧ ¬q are logically equivalent by using a series of logical equivalences. ¬(p v (¬ p ∧ q)) = ¬p ∧ ¬ (¬p ∧ q) = ¬p ∧ [¬ (¬ p) v ¬q] = ¬p ∧ (p v ¬q) = (¬p ∧ p) v (¬p ∧ ¬q) = F v (¬p ∧ ¬q) = (¬p ∧ ¬q) v F = ¬p ∧ ¬q

De Morgan Law De Morgan Law Double Negation Law Distributive Law Useful Commutative Law Identity Law

4. Determine whether (¬p ∧ (¬p ∧ q)) → q is a tautology.

(¬p ∧ (¬p ∧ q)) → q ¬ (¬p ∧ (¬p ∧ q)) v q p v ¬ (¬p ∧ q)) v q p v ( p v ¬q) v q (p v p) v (¬q v q ) pvT T

Useful De Morgan Law and Double Negation Law De Morgan Law Associative Law Idempotent Law and Useful Domination Law...