“Simplified” is not “minimal”: what a simplifier can promise
The simplifier found no law that shortens (A AND B) OR (A AND NOT B) OR (NOT A AND B), yet the function is A OR B. “Simplified” means no rewriting law applies; it does not mean no shorter form exists, and the engine says exactly that.
The result below is honest about its limits. The simplifier applies named laws one at a time, each one a local rewrite that makes the expression smaller by its measure; none of its laws matches this three-term expression, so it reports it unchanged, with the note that smaller equivalent forms are not ruled out. The Karnaugh map, which looks at the whole function rather than the syntax, derives A OR B and checks it equivalent on every row, and the conjunctive normal form converter reaches A OR B as well.
The gap is structural. Merging the first two terms needs the complement law applied to B OR NOT B after factoring A out, and the simplifier does not factor; absorbing the third needs the consensus theorem, which is not a local rewrite either. A method that enumerates the rows, or selects prime implicants, sees the function as a whole, which is why the map finds what the laws miss.
This is why the engine labels its result simplified rather than smallest, and why the semantics specification never claims a form to be the shortest. The hub page for tools says which tool to use when: the simplifier for a readable, step-by-step reduction with each law named; the Karnaugh map for an irreducible sum of products on up to four variables; the normal-form converter for a canonical form.
f := (A AND B) OR (A AND NOT B) OR (NOT A AND B) · output Truth table
| # | A | B | f |
|---|---|---|---|
| 0 | FALSE | FALSE | FALSE |
| 1 | FALSE | TRUE | TRUE |
| 2 | TRUE | FALSE | TRUE |
| 3 | TRUE | TRUE | TRUE |
Simplification, step by step
The engine found no law that shortens this expression; it is already as short as its rewriting laws can make it.
(A AND B) OR (A AND NOT B) OR (NOT A AND B)Karnaugh map
- cells m2, m3
A - cells m1, m3
B
Normal forms
Conjunctive normal form (CNF)
A OR B1 clauses, 2 literals · complete · Distribution ×2, Complement ×2, Absorption ×2
Disjunctive normal form (DNF)
(A AND B) OR (A AND NOT B) OR (NOT A AND B)3 terms, 6 literals · complete
Plain-language reading
Beginnerf is true exactly when at least one of the following holds (several may hold):
1. both A and B are true
2. A is true and B is false
3. A is false and B is true
In every other case, f is false.
About the operators:
- OR is inclusive: it is true when at least one part is true, including when several are.
- AND is true only when all of its parts are true.
- NOT reverses a value: true becomes false and false becomes true.
Intermediatef is true exactly when at least one of the following holds (several may hold):
1. both A and B are true
2. A is true and B is false
3. A is false and B is true
In every other case, f is false.
Advancedf is true exactly when at least one of the following holds (several may hold) [inclusive disjunction ∨]:
1. both A and B are true [conjunction ∧]
2. A is true and B is false [conjunction ∧]
3. A is false and B is true [conjunction ∧]
In every other case, f is false.
Formal notation: (A ∧ B) ∨ (A ∧ ¬B) ∨ (¬A ∧ B)
The common slip
Reading “no law applies” as “this is as short as it gets”. The simplifier's own summary says that smaller equivalent forms are not ruled out; when the form matters, run the map or the converter and compare.
Questions
Why not make the simplifier try harder?
It could, and it would then stop being a step-by-step explanation. Its value is that every step is a named law a student can check; the map and the prime-implicant method are the tools for the whole-function question, and they are one click away.
Does the Karnaugh map find the smallest form?
It derives a cover with no removable term and checks it exactly on every row. The engine calls that irreducible and does not claim it is the smallest possible, because for more than a few variables that claim would need a different kind of search.
Related
Tools
- Logical equivalence checker: two expressions, exact verdict, counterexample
- Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns
Practice
- Boolean simplification exercises with answers: basic
- Boolean simplification exercises with answers: intermediate
- Boolean simplification exercises with answers: advanced
See also
- Boolean algebra calculator and simplifier with steps (every law named and verified)
- Karnaugh map solver: 2 to 4 variables, groups, SOP and POS, don’t-cares from assumptions
- CNF and DNF converter: conjunctive and disjunctive normal forms, minterms and maxterms
- Consensus theorem: AB + A'C + BC = AB + A'C, with proof
- Karnaugh map grouping rule: adjacent cells drop the variable that changes
- Lesson 7: Simplification laws (intermediate, 12 min)
- Karnaugh map vs Quine–McCluskey
Pitfalls
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.