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
|