Principle of distributivity
The principle of distributivity states that the algebraic distributive law is valid, where both logical conjunction and logical disjunction are distributive over each other.
In Propositional Logic
For any propositions A, B and C, the following equivalences hold:Proof using truth tables
The distributive laws can be verified using truth tables.Conjunction distributes over disjunction
For the equivalence, the truth table is:| A | B | C | B ∨ C | A ∧ | A ∧ B | A ∧ C | ∨ |
| T | T | T | T | T | T | T | T |
| T | T | F | T | T | T | F | T |
| T | F | T | T | T | F | T | T |
| T | F | F | F | F | F | F | F |
| F | T | T | T | F | F | F | F |
| F | T | F | T | F | F | F | F |
| F | F | T | T | F | F | F | F |
| F | F | F | F | F | F | F | F |
As seen from the table, the columns for and are identical. Therefore, the equivalence is valid.
Disjunction distributes over conjunction
For the equivalence, the truth table is:| A | B | C | B ∧ C | A ∨ | A ∨ B | A ∨ C | ∧ |
| T | T | T | T | T | T | T | T |
| T | T | F | F | T | T | T | T |
| T | F | T | F | T | T | T | T |
| T | F | F | F | T | T | T | T |
| F | T | T | T | T | T | T | T |
| F | T | F | F | F | T | F | F |
| F | F | T | F | F | F | T | F |
| F | F | F | F | F | F | F | F |
As seen from the table, the columns for and are identical. Therefore, the equivalence is valid.