How do I make Boolean absorption laws click?

Try a truth table when a rule feels suspicious. For A AND (A OR B), there are only four combinations of A and B. If A is false, the whole expression is false. If A is true, A OR B is already true, so the whole expression is true.

Either way, the result matches A. That shows why A AND (A OR B) simplifies to A, regardless of B. You can check A OR (A AND B) the same way.

Think of absorption as removing a condition that can't change the result. In A OR (A AND B), the second part can only be true when A is already true. So it never adds a case that A didn't already cover.

The other form, A AND (A OR B), says A must be true, and then also asks for A OR B. But once A is true, that second requirement is automatically satisfied.

The notation is a bit of a trap. Boolean multiplication behaves like AND, but Boolean addition means OR, not ordinary addition. In particular, A + A = A, not 2A, and 1 + B = 1.

With that in mind, factoring gives A + AB = A(1 + B) = A. You can still use algebraic manipulation, but the rules for the operators aren't all the same as arithmetic.

For practice, look for a standalone term alongside a larger term containing it. In A + BC + B, the standalone B absorbs BC, leaving A + B. You don't need to factor the whole expression to spot it.

After each step, check a few inputs or make a full truth table. A counterexample is enough to disprove a proposed simplification, while matching every possible input verifies equivalence.