2. Logical Equivalence
Logical equivalence is explored through the examination of propositional logic, including the definitions of tautology, contradiction, and contingency. The chapter emphasizes the significance of the contrapositive and biconditional statements and introduces standard logical identities, such as De Morgan's laws and the distributive laws. Techniques for simplifying complex logical expressions using known identities are also discussed, providing a foundation for proving logical equivalence without relying solely on truth tables.
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What we have learnt
- Logical equivalence occurs when two compound propositions yield the same truth values.
- Tautologies are statements that are always true, while contradictions are always false; contingencies can be either.
- Standard logical identities can be applied to simplify complex expressions and prove logical equivalence.
Key Concepts
- -- Logical Equivalence
- Two compound propositions are logically equivalent if they have the same truth values across all scenarios.
- -- Tautology
- A proposition that is always true, regardless of the truth values of its variables.
- -- Contradiction
- A proposition that is always false, no matter the truth values assigned to its variables.
- -- Contingency
- A proposition that can be true in some cases and false in others, not classified as a tautology or contradiction.
- -- Biconditional
- A logical statement of the form 'p if and only if q' indicating that both propositions are equivalent.
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