G4G0-2/AI & Data Mining/Week 21/Chapter 21 - Transformational Proofs.md

1. George Boole
2. Truth Tables for a) Negation b) Contraposition

a) Negation Law ¬¬p ≡ p

p	¬p	¬(¬p)	¬(¬p) ⇔ p
T	F	T	T
F	T	F	T

b)

Contraposition Law p ⇒ q ≡ ¬q ⇒ ¬p

p	¬p	q	¬q	p ⇒ q	¬q ⇒ ¬p	p ⇒ q ⇔ ¬q ⇒ ¬p
T	F	T	F	T	T	T
T	F	F	T	F	F	T
F	T	T	F	T	T	T
F	T	F	T	T	T	T

p ⇒ q ⇔ ¬q ⇒ ¬p MUST be true, since p ⇒ q and ¬q ⇒ ¬p are shown in the truth table to be the same logical equivalence

1. Provide names of laws
   1. Negation Law
   2. De Morgan's Law
   3. Negation Law
   4. De Morgan's Law
   5. Negation Law Twice
   6. Associative Law
   7. De Morgan's Law
   8. De Morgan's Law
   9. Negation Law Twice
2. Show logical equivalence p ⇒ q ¬q ⇒ ¬p p ⇒ q ≡ (¬p) v q ≡ q v (¬p) ≡ ¬ (¬q) v (¬p) ≡ (¬q) ⇒ (¬p)