Boolean Laws & Simplification

Simplify Boolean expressions using the laws of Boolean algebra — any equivalent form is accepted.

A Boolean expression is built from variables that are only ever 0 or 1, combined with AND, OR, and NOT. Two expressions can look completely different and still be the same circuit — same output for every possible input. Simplification means finding a shorter expression with that exact same behavior, not a different one that merely resembles it.

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Because of that, there's no single "correct" simplified answer to memorize the shape of. Any expression that matches the original on every row of the truth table is correct, even if it doesn't look like the form your textbook lands on.

The laws that let you rewrite an expression without changing what it computes:

\begin{aligned} A + AB &= A \\ A(B+C) &= AB + AC \\ \overline{AB} &= \overline{A} + \overline{B} \\ \overline{A+B} &= \overline{A}\,\overline{B} \\ A + \overline{A} &= 1 \end{aligned}

Absorption, Distributive, De Morgan's, Complement

$\overline{A}$ the complement (NOT) of A — 1 when A is 0, and 0 when A is 1.
$AB$ A AND B, written as plain juxtaposition — no separate multiplication symbol.
$A + B$ A OR B.

De Morgan's is the one that trips people up: complementing a whole expression flips every AND to an OR (and vice versa) AND complements every variable inside it. Do both, every time — flipping only the operator, or only the variables, gives you something that isn't equivalent at all.

The fastest way to check any proposed simplification, by hand or in your head, is the truth table: list every combination of the inputs, evaluate both the original and your simplified expression on each one, and confirm they match on all of them. Two expressions that agree on every row are the same expression, full stop — that's what "logically equivalent" means, and it's exactly what this site checks your answer against.

Worked example

\overline{AB}

$\overline{AB} = \overline{A} + \overline{B}$
Apply De Morgan's law (AND form): complementing a product complements each factor and swaps AND for OR.
$\begin{array}{cc|c|c} A & B & \overline{AB} & \overline{A}+\overline{B} \\ \hline 0&0&1&1 \\ 0&1&1&1 \\ 1&0&1&1 \\ 1&1&0&0 \end{array}$
Verify on the truth table: both columns agree on all 4 rows, so the rewrite is valid.

Answer:

\overline{A} + \overline{B}

How to solve one of these

1.
Write out the original expression. Note which variables it uses — that fixes how many rows a truth-table check needs ( $2^n$ for $n$ variables).
2.
Look for a law that applies. Absorption ( $A + AB = A$ ), distribution, De Morgan's, or a repeated/complemented term are the usual starting points.
3.
Apply one law at a time. Rewrite the expression, don't try to jump straight to the final simplified form in one step.
4.
Repeat until nothing more simplifies. Keep applying laws as long as the expression keeps getting shorter.
5.
Check by truth table if unsure. Evaluate the original and your result on every input combination — they must match on all of them, not just the ones you checked by eye.

\overline{A}, \quad AB, \quad A + B

NOT A, A AND B, A OR B — the three operations every Boolean expression is built from.

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