LECTURE 2: PROPOSITIONAL EQUIVALENCES
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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.
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The document discusses mathematical logic and reasoning. It notes that mathematics uses deductive reasoning, where conclusions are logically derived from accepted statements. The mathematician uses logic to draw conclusions about any imaginable mathematical structure. Logic is important in other areas like computer programming, where algorithms are constructed similarly to mathematical proofs. The basics of propositional logic are then introduced, including logical connectives like negation, conjunction, disjunction, implication, and equivalence. Truth tables are used to evaluate propositions composed of connectives. Different types of propositions like tautologies and contradictions are defined. Rules of replacement and valid rules of inference in deductive arguments are also covered.
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LECTURE 2: PROPOSITIONAL EQUIVALENCES
- 1. WENNILOUPORAZO 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
- 21. Show that ¬(p → q) and p ∧ ¬q are logically equivalent.example1: ZimCore Hubs • DECEMBER, 2020 Solution: ¬(p → q) ≡ ¬(¬p ∨ q) by previous example ≡ ¬(¬p) ∧ ¬ q De Morgan's Law ≡ p ∧ ¬ q Double Negation Law