Boolean logic lessons: from NOT and AND to Karnaugh maps

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.

Each lesson is a sequence of short steps. A step opens a small model (two or three rules over a few inputs), explains one idea, and asks you to try one thing. The text makes claims: that two formulas are equivalent, that a rule is a tautology, that an argument is invalid. The engine checked all 135 claims and worked examples of these lessons when this site was built; a lesson whose claim failed would not have been published.

These pages are the readable version. The interactive version in the app lets you edit the model, toggle inputs, run the checks yourself and solve the 52 exercises with immediate feedback. Both are free and need no account.

The path runs from the operators and XOR through grouping, truth tables and conditions (if, only if, unless) to De Morgan, simplification, arguments and counterexamples, and ends with normal forms, Karnaugh maps and designing tests that find bugs. Two further lessons go beyond TRUE and FALSE: category inputs, ordered scales and counting constraints, and reading a result backwards through necessary conditions, ways to qualify, smallest changes and strictness.

• Lesson 1: NOT, AND, OR: the basic operators (beginner, 10 min)
  The first lesson builds the entry rule (member OR invited) AND NOT banned one operator at a time: AND, then inclusive OR, then NOT.
• Lesson 2: XOR: exclusive OR and parity (beginner, 10 min)
  Exclusive or in three steps: how A XOR B differs from A OR B, why a stairwell light is downstairs XOR upstairs, and why A XOR B XOR C means odd parity rather than “exactly one”.
• Lesson 3: Grouping and precedence (beginner, 12 min)
  Which operator applies first when there are no brackets, how the parser reads A OR B AND C, and why dropping the brackets from the entry rule lets a banned member in.
• Lesson 4: Truth tables (beginner, 12 min)
  A truth table lists every assignment of the inputs in a fixed order and the output on each.
• Lesson 5: If, only if, unless: necessary and sufficient conditions (intermediate, 15 min)
  “If”, “only if”, “if and only if” and “unless” each translate to an implication that points a particular way.
• Lesson 6: De Morgan’s laws (intermediate, 12 min)
  How to move a NOT through AND and OR without changing the meaning: negating a conjunction gives a disjunction of negations, and the other way round.
• Lesson 7: Simplification laws (intermediate, 12 min)
  Six laws that make formulas shorter, identity, domination, idempotence, complement, absorption and double negation, applied one at a time with the exact checker confirming each step.
• Lesson 8: Arguments and validity (intermediate, 15 min)
  State premises and a conclusion, and ask whether the conclusion follows.
• Lesson 9: Counterexamples (intermediate, 12 min)
  One concrete assignment can refute an equivalence or an argument.
• Lesson 10: Normal forms: DNF, CNF, minterms, and maxterms (advanced, 15 min)
  Any formula can be rewritten as an OR of AND-terms (DNF) or an AND of OR-clauses (CNF).
• Lesson 11: Karnaugh maps (advanced, 15 min)
  A Karnaugh map is a truth table folded so that neighbouring cells differ in one input.
• Lesson 12: Designing tests that find bugs (advanced, 15 min)
  Tests that find bugs need expected results taken from the requirement, scenarios that show each input matters, and a way to check whether the suite would notice a mistake.
• Lesson 13: Categories, scales, and counting (intermediate, 20 min)
  Rules say “the plan is one of these”, “rated BBB- or better” and “at least two of three criteria”.
• Lesson 14: Why this result: reasons and strictness (advanced, 20 min)
  A verdict says whether a rule holds; this lesson reads the result backwards.

Questions

Do I need to follow the lessons in order?

The order is a learning path, but every lesson stands on its own and names what it assumes. Someone who already knows truth tables can start at De Morgan or at the arguments lesson.

Are the exercises solved on these pages?

No. The pages list the exercises so you know what each lesson practises; the solutions are checked in the app, where every answer gets exact feedback from the same engine.

Related

Guides

• What is discrete math?

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. 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.