A·B + Ā·C + B·C = A·B + Ā·C — Boolean algebra laws
introconsensus
Answer
verified equivalent (both sides → A′ · C + A · B)
Why this example is worth doing
The consensus theorem, stated with its usual pattern: two terms containing a variable and its complement, and a third built from the remaining literals. The page shows why the third is implied — whatever A does, one of the first two terms already covers every row where B·C holds — and then makes the practical qualification that redundant does not mean useless. The consensus term is exactly the one you add back to eliminate a static-1 hazard in a real gate network, so this law is read in both directions.
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 + A′ · C + B · CA · 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 | 1 |
| 2 | 0 | 1 | 0 | 0 |
| 3 | 0 | 1 | 1 | 1 |
| 4 | 1 | 0 | 0 | 0 |
| 5 | 1 | 0 | 1 | 0 |
| 6 | 1 | 1 | 0 | 1 |
| 7 | 1 | 1 | 1 | 1 |