Associative laws
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Associative laws
The associative laws state that when combining three or more propositions with conjunction or disjunction, the grouping of the operations does not affect the truth value.
Statements
- (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
- (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
Explanation
Parentheses can be rearranged without changing the logical meaning.
Examples
- "((I study AND I practice) AND I succeed)" is equivalent to "(I study AND (I practice AND I succeed))".
- "((It rains OR it snows) OR it is windy)" is equivalent to "(It rains OR (it snows OR it is windy))".
Truth Table (Conjunction)
| p | q | r | (p ∧ q) ∧ r | p ∧ (q ∧ r) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | T | F | F | F |
| T | F | T | F | F |
| T | F | F | F | F |
| F | T | T | F | F |
| F | T | F | F | F |
| F | F | T | F | F |
| F | F | F | F | F |
Truth Table (Disjunction)
| p | q | r | (p ∨ q) ∨ r | p ∨ (q ∨ r) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | T | F | T | T |
| T | F | T | T | T |
| T | F | F | T | T |
| F | T | T | T | T |
| F | T | F | T | T |
| F | F | T | T | T |
| F | F | F | F | F |