An algebraic proof transforms one side of an equation into the other using only the laws of boolean algebra. Each line is equal to the one before, so the first and last lines are equal.
Good habits:
- One law per step, and write its name next to the step.
- Work from the more complicated side toward the simpler one; it is easier to remove things than to invent them.
- Keep brackets until a law tells you to remove them.
- Use reverse steps when needed. Sometimes you must make an expression bigger first, for example writing A as (identity) so you can factor.
Compared with perfect induction, algebra scales to any number of variables and shows why something is true. Its risk is a wrong step, so a quick row check at the end is wise.
Proofs are also how new rules are justified. The absorption law, the redundant literal rule and the consensus theorem can all be proved from the basic laws.