Enter a boolean expression like A'B + AB + AB'C, or a list of minterms with optional don't-cares. The simplifier finds the minimal sum of products and product of sums with the Quine-McCluskey method and shows every step.
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Input type
' or ! or ~ or NOT = NOT, juxtaposition · * & AND = AND, + | OR = OR, ^ XOR, -> implies. Brackets allowed.
Ranges allowed: 0-3, 7
Enter your values to see the result.
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How to use the Boolean Algebra Simplifier
Pick the input type: a boolean expression, or a minterm list with variable names.
Type the expression, e.g. A'B + AB + AB'C (' means NOT, juxtaposition means AND, + means OR), or the minterm numbers such as 0, 2, 5-7.
Add don't-cares if some input combinations can never happen.
Press Simplify to get the minimal SOP and POS with their term and literal counts.
Follow the steps and the prime implicant chart to see why the answer is minimal, then copy it.
Boolean algebra laws used in simplification
Law
AND form
OR form
Identity
A · 1 = A
A + 0 = A
Null (annulment)
A · 0 = 0
A + 1 = 1
Idempotent
A · A = A
A + A = A
Complement
A · A' = 0
A + A' = 1
Double negation
(A')' = A
Absorption
A(A + B) = A
A + AB = A
Combining (adjacency)
(A + B)(A + B') = A
AB + AB' = A
Redundant literal
A(A' + B) = AB
A + A'B = A + B
Distributive
A(B + C) = AB + AC
A + BC = (A + B)(A + C)
De Morgan
(AB)' = A' + B'
(A + B)' = A'B'
Consensus
(A + B)(A' + C)(B + C) = (A + B)(A' + C)
AB + A'C + BC = AB + A'C
Worked example: A'B + AB + AB'C
The expression is 1 for minterms 2, 3, 5, 6 and 7 (binary 010, 011, 101, 110, 111).
Combining neighbours gives 01- (A'B), -1- after a second round (B), and 1-1 (AC).
The prime implicants are B (covers 2, 3, 6, 7) and AC (covers 5, 7). Both are essential: only B covers 2 and only AC covers 5.
Minimal SOP: F = B + AC, down from 7 literals to 3. The minimal POS is (A + B)(B + C).
The same answer follows algebraically: A'B + AB = B (combining), and B + AB'C = B + AC (redundant literal, because B + B'(AC) = B + AC).
Tips
Write products without a symbol. With single-letter variables on, AB'C means A AND NOT B AND C. Turn it off for multi-letter names such as start AND NOT stop.
Compare SOP and POS. If the function has many 1s, the POS (built from the 0s) is often the shorter one.
XOR does not shrink. Parity functions like A ⊕ B ⊕ C have no adjacent minterms, so their two-level form stays large; keep them as XOR gates.
Need the full table? The Truth Table Generator shows every row, minterms and the canonical DNF/CNF of any expression.
Frequently asked questions
How do I simplify a boolean expression?
By hand you apply laws such as absorption (A + AB = A), combining (AB + AB' = A) and consensus (AB + A'C + BC = AB + A'C), or you draw a Karnaugh map. Those approaches depend on spotting the right move. The Quine-McCluskey method used here is systematic: it lists all prime implicants and then picks the smallest set that covers every minterm, so the result is guaranteed minimal.
What is the Quine-McCluskey method?
It is a tabular algorithm for two-level logic minimisation. Minterms are grouped by their number of 1 bits, and terms in neighbouring groups that differ in exactly one bit are merged, replacing that bit with a dash. Terms that cannot merge any further are prime implicants. A prime implicant chart then selects the essential primes and the cheapest cover for the remaining minterms.
What is the difference between SOP and POS?
Sum of products (SOP) is an OR of AND terms, such as AB' + C, and maps directly to AND-OR or NAND-NAND logic. Product of sums (POS) is an AND of OR clauses, such as (A + C)(B' + C), and maps to OR-AND or NOR-NOR logic. The minimal POS is found by minimising the zeros of the function and applying De Morgan's law, and it can be smaller or larger than the minimal SOP.
What are don't-care terms?
Don't-cares are input combinations that never occur or whose output does not matter, written d(…) or X. For example, BCD digits only use 0 to 9, so inputs 10 to 15 of a 4-bit decoder are don't-cares. The simplifier may treat each one as 0 or 1, whichever gives a smaller answer, which often removes whole terms.
Why can there be more than one minimal answer?
Some functions have several covers with the same number of terms and literals. A classic example is Σm(0, 1, 2, 5, 6, 7), which is both A'B' + BC' + AC and A'C' + B'C + AB. The tool lists the alternatives it finds so you can pick the one that shares gates with the rest of your circuit.
How is the minterm number related to the variables?
Variables are taken in the order shown (alphabetical for expressions), and the first one is the most significant bit. With A, B, C, D the minterm 11 is binary 1011, i.e. A = 1, B = 0, C = 1, D = 1, which is the product AB'CD.
Is this the same as a Karnaugh map?
It gives the same minimal answer as a correctly drawn Karnaugh map, but it works for more than four variables and does not depend on spotting the right groups. The prime implicants correspond to the largest possible K-map groups and the essential ones are the groups that cover a cell no other group covers.
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