The distributive law is the one that mixes AND with OR. It has two forms.
1. AND over OR: =
This is ordinary "multiplying out brackets". Read right to left, it is factoring: pulling a shared literal out of several terms.
2. OR over AND: =
This one has no match in ordinary arithmetic (2 + 3 × 4 is not 5 × 6), so it surprises people. It is true in Boolean algebra: if A = 1 both sides are 1, and if A = 0 both sides reduce to . See or over and.
Why it matters:
- Multiplying out turns any expression into a sum of products, the standard starting point for simplification.
- Factoring is the first move of combining terms and of many proofs, such as = .
- The OR-over-AND form turns an SOP into a product of sums, and collapses brackets like in one step.
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 |