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

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AI & Data Mining/Week 21/Week 21 - Transformational Proofs in Propositional Logic.md
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2. Prove De Morgan's laws using truth tables or transformational proofs.
3. Prove the laws involving true and false using truth tables or transformational proofs.
4. Prove the laws of simplification using truth tables or transformational proofs.
5. Prove the equivalence of two given formulae using transformational proofs (as demonstrated in slides 21 and 22).
5. Prove the equivalence of two given formulae using transformational proofs (as demonstrated in slides 21 and 22).