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

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+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)