A + B·C = (A + B)·(A + C) — Boolean algebra laws
coredistributivity of OR over AND
Answer
verified equivalent (both sides → B · C + A)
Why this example is worth doing
The distributive law students do not believe, because its arithmetic analogue is false: a + bc is not (a+b)(a+c) in ordinary numbers. In Boolean algebra both distributions hold, and this is the direction that converts a sum of products into a product of sums by hand. The page contrasts it explicitly with integer arithmetic, since the transfer of intuition from arithmetic is the actual source of the error, and notes that this identity is what makes SOP and POS interconvertible without a truth table.
Try your own input in the Boolean laws & theorems. Every law, both duals, each with a truth-table proof you can check.
How the answer is reached
The law
A + B · C(A + B) · (A + C)— Both sides have the same truth table over the union of their variables.
Truth table
| # | A | B | C | F |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 | 0 |
| 2 | 0 | 1 | 0 | 0 |
| 3 | 0 | 1 | 1 | 1 |
| 4 | 1 | 0 | 0 | 1 |
| 5 | 1 | 0 | 1 | 1 |
| 6 | 1 | 1 | 0 | 1 |
| 7 | 1 | 1 | 1 | 1 |