Boolean logic reference: laws, gates, truth tables, K-maps and free tools
A reference on two-valued Boolean algebra and propositional logic where every table, map, simplification and verdict was computed by the Tangibool engine when the site was built, and every claim was re-checked by its exact checker.
Most pages about Boolean logic tell you what a law says. These pages show it: each law carries a truth-table proof over every assignment, each worked example carries its simplification steps with the law named at every step, and each equivalence or validity claim carries the engine's verdict, the method it used, and a counterexample when one exists. Nothing here was typed by hand and then trusted; the generator fails the build if an identity does not hold.
Start with the free tools if you have an expression in front of you: the truth table generator, the simplifier with named steps, the Karnaugh map solver, the CNF and DNF converter, the equivalence checker and the argument validity checker all run in your browser, and nothing you type leaves it. Start with the laws, operators and lessons if you are learning; with the pitfalls, testing pages and language pages if you write rules for a living.
Limits are stated where they apply. Truth tables stop at 12 inputs, Karnaugh maps at 4 effective variables, and a simplifier result is labelled simplified, not minimal. A result the engine could not determine within its budget says so rather than guessing.
Laws
Every law of Boolean algebra the engine applies, each stated in four notations and proved on every row of its truth table.
32 pages
Operators and gates
The logic gates and operators the engine understands: truth tables for two, three and four inputs, every accepted spelling, and the precedence that decides how an unbracketed expression is read.
11 pages
Tools
Nine free tools built on the Tangibool engine: they run in your browser, state their limits, and show their working.
9 pages
Lessons
The fourteen guided lessons of the Tangibool app, from NOT and AND to Karnaugh maps, counting and strictness, with every example and claim verified at build time.
14 pages
Glossary
Sixty terms of Boolean algebra, propositional logic and rule testing, each defined in its first sentence, with a computed example and its Spanish equivalent.
60 pages
Worked examples
Sixty worked examples, from the half adder and the majority function to eligibility rules and feature flags, each with its truth table, Karnaugh map, simplification steps, normal forms, gate forms, diagram, plain-language reading and code, all computed and verified.
60 pages
Pitfalls
Twenty-five ways a Boolean rule goes wrong, each with the wrong and the right formula side by side, the row where they differ, and the fix.
25 pages
Testing and rules
How to test a decision rule: decision tables, test cases from a truth table, MC/DC with the engine's independence pairs, condition versus decision coverage, mutation testing, and the patterns behind eligibility, access-control and feature-flag rules.
15 pages
Languages
Boolean logic in Python, JavaScript, SQL, Excel, Java, C, SMT-LIB and LaTeX: the operators, the precedence, the traps, and the same rule exported by the engine where it has an exporter.
9 pages
Compare
Side-by-side comparisons that keep coming up: sum of products against product of sums, CNF against DNF, the Karnaugh map against Quine–McCluskey, truth tables against decision tables, NAND against NOR, and the free simplifier tools measured on the same expressions.
10 pages
Practice
Practice sets with solutions verified by the engine: truth tables, simplification, Karnaugh maps, normal forms, argument validity, De Morgan, translation from English, gates and circuits, and the sets for philosophy courses, ISTQB decision tables and MC/DC.
20 pages
Questions
What does “checked by the engine” mean on these pages?
The Tangibool engine (the same code that runs the app) built every truth table by enumerating every assignment, proved every law and every simplification step with an exact equivalence check, and replayed every counterexample through an independent evaluator before it was written to the page. The build date and the semantics version are printed at the foot of every page.
Is anything here generated by AI?
No mathematical result on these pages comes from a language model. The explanations were written by people and reviewed; the tables, maps, steps and verdicts were computed. Where a page says a formula is equivalent to another, it is because the checker found no row where they differ.
Can I use the tools without an account?
Yes. The tools run entirely in your browser, need no account, store nothing and send nothing. The “Open in Tangibool” button takes the same rule into the full workspace, which is also free to use signed out.
Related
Guides
Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build . Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.