Logical Equivalence
When two logical formulas have identical truth values in all cases
P ≡ Q means P and Q are logically equivalentLogical equivalence (≡) means two formulas always have the same truth value. No matter what truth values we assign to the variables, equivalent formulas will always produce the same result. This is the foundation for logical simplification and transformation.
Key Characteristics:
- • Same truth value in all possible cases
- • Can be verified by truth tables
- • Forms the basis for logical transformations
- • Enables formula simplification
Common Examples:
P ∧ Q ≡ Q ∧ P (commutativity), ¬(P ∧ Q) ≡ ¬P ∨ ¬Q (De Morgan's), and P → Q ≡ ¬P ∨ Q (material implication).
P ≡ Q ↔ (P and Q have identical truth tables)Basic Equivalence
Formula 1:
Formula 2:
Equivalence:
De Morgan's Equivalence
Original:
Equivalent:
Law:
Material Implication
Implication:
Disjunctive Form:
Equivalence:
Key Point: Logical equivalence allows us to transform formulas while preserving their meaning, enabling simplification and standardization.
Example 1: Commutative Laws(Order independence)
Conjunction:
Disjunction:
Explanation: The order of operands doesn't matter for AND and OR operations. "Sunny and warm" means the same as "warm and sunny".
Example 2: Double Negation(Cancellation)
Double Negative:
Simplified:
Explanation: Two negations cancel out: "It's not true that it's not raining" is equivalent to "It's raining".
Example 3: Absorption Law(Simplification)
Complex Formula:
Simplified:
Explanation: If P is true, then P ∨ Q is also true, making P ∧ (P ∨ Q) just P. If P is false, the whole expression is false.