Distributive laws
The distributive laws describe how conjunction and disjunction distribute over each other.
They show that a conjunction can be distributed over a disjunction, and a disjunction can be distributed over a conjunction.
Statements
- p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
- p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
Explanation
These rules are similar to the distributive property in arithmetic.
They allow logical formulas to be rewritten in different but equivalent forms.
Examples
- "I study AND (I pass OR I fail)" is equivalent to "(I study AND I pass) OR (I study AND I fail)".
- "I travel OR (I save money AND I rest)" is equivalent to "(I travel OR I save money) AND (I travel OR I rest)".
Truth Table (First Law)
| p |
q |
r |
p ∧ (q ∨ r) |
(p ∧ q) ∨ (p ∧ r)
|
| T |
T |
T |
T |
T
|
| T |
T |
F |
T |
T
|
| T |
F |
T |
T |
T
|
| 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 (Second Law)
| p |
q |
r |
p ∨ (q ∧ r) |
(p ∨ q) ∧ (p ∨ 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 |
F |
F
|
| F |
F |
T |
F |
F
|
| F |
F |
F |
F |
F
|