LECTURE 2: PROPOSITIONAL EQUIVALENCES

An important type of step used in a mathematical argument is the replacement of a statement with another with the same truth value. Because of this, methods that propositions with the same truth value as a given compound proposition are used extensively in the construction of mathematical arguments.

1. WENNILOU PORAZO DISCRETE MATHEMATICS LECTURE 2 PROPOSITIONAL EQUIVALENCES MLG COLLEGE OF LEARNING

2. Tautology Definition A compound proposition that is always true , no matter what the truth values of the propositional variables that occur in it, is called a tautology. discrete mathematics

3. contradiction Definition A compound proposition that is always false. discrete mathematics

4. contingency Definition A compound proposition that is neither a tautology nor a contradiction. discrete mathematics

5. complete the table: DECEMBER 2020 mlg college of learning discrete mathematics p ∨ ¬p

6. problem 1: DECEMBER 2020 mlg college of learning discrete mathematics p ∨ ¬p

7. solution: DECEMBER 2020 mlg college of learning discrete mathematics p ∨ ¬p T T T T Therefore, p ∨ ¬p is a Tautology .

8. problem 2: DECEMBER 2020 mlg college of learning discrete mathematics p ∧ ¬p

9. solution: DECEMBER 2020 mlg college of learning discrete mathematics p ∧ ¬p F F F F Therefore, p ∧ ¬p is a Contradiction .

10. logical equivalences discrete mathematics Compound propositions that have the same truth values in all possible cases are logically equivalent. The compound prepositions p and q are called logically equivalent if p ↔ q is a tautology. The notation p ≡ q denotes that p and q are logically equivalent.

11. NOTE: The symbol ≡ is not a logical connective, and p ≡ q is not a compound proposition but rather is the statement that p ↔ q is a tautology. discrete mathematics The symbol ⇔ is sometimes used instead ≡ of to denote logical equivalence.

12. ALSO: The compound prepositions p and q are equivalent if and only if the columns giving their truth values agree. discrete mathematics

13. ¬(P ∧ Q) ≡ ¬P ∨ ¬Q DE MORGAN'S LAWS DISCRETE MATHEMATICS ¬(P ∨ Q) ≡ ¬P ∧ ¬Q

14. problem 1: september 2020 mlg college of learning discrete mathematics Show that ¬(p ∧ q) and ¬(p ∨ q) are logically equivalent.

15. solution: DECEMBER 2020 mlg college of learning discrete mathematics Using a truth table

16. explanation: DECEMBER 2020 mlg college of learning discrete mathematics Because the truth values of the compound propositions ¬(p ∨ q) and ¬(p ∧ q) agree for all possible combinations of the truth values of p and q, it follows that ¬(p ∨ q) ↔ ¬(p ∧ q) is a tautology and that these compound propositions are logically equivalent.

17. Proof that ¬(p ∨ q) ↔ ¬(p ∧ q) is a tautology DECEMBER 2020 mlg college of learning discrete mathematics

18. LOGICAL EQUIVALENCES DISCRETE MATHEMATICS

19. LOGICAL EQUIVALENCES DISCRETE MATHEMATICS

20. LOGICAL EQUIVALENCES DISCRETE MATHEMATICS

21. Show that ¬(p → q) and p ∧ ¬q are logically equivalent. example 1: ZimCore Hubs • DECEMBER, 2020 Solution: ¬(p → q) ≡ ¬(¬p ∨ q) by previous example ≡ ¬(¬p) ∧ ¬ q De Morgan's Law ≡ p ∧ ¬ q Double Negation Law