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Boolean Algebra Simplifier

Simplify Boolean expressions with algebraic laws, Quine-McCluskey and Karnaugh maps using MiniWebtool’s Boolean Algebra Simplifier. Enter any AND, OR, NOT, XOR, NAND or NOR expression to get the minimal SOP/POS form, truth table, K-map, gate diagram and NAND-only equivalents.

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Boolean Algebra SimplifierTry it now — free ▼
On a Karnaugh map, a pair of adjacent 1s drops one variable

Enter a Boolean expression below to get the minimal form, the truth table, the Karnaugh map and the steps.

Examples:
Insert:
SYNTAX ACCEPTED
AND: & · AB  |  OR: + |  |  NOT: ~ ! A'  |  XOR NAND NOR  |  or a list: F(A,B,C,D) = Σm(1,3,7) + d(0,2)

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About Boolean Algebra Simplifier

The Boolean Algebra Simplifier reduces any logic expression to its minimal form using the Quine-McCluskey algorithm and Karnaugh map analysis. Enter expressions with AND, OR, NOT, XOR, NAND, NOR (or symbols like &, |, ~, ^, postfix A'), and instantly receive the full truth table, a colored K-map with prime implicant loops, a prime implicant chart marking essentials, a step-by-step derivation, and equivalent NAND-only and NOR-only implementations.

What the Simplifier Produces

📊
Truth Table
All \(2^n\) input combinations with the expression's output.
🗺️
Karnaugh Map
Gray-code grid with each prime implicant outlined.
🎯
Minimal SOP / POS
Fewest literals, fewest terms — the optimal forms.
🧩
Prime Implicants
With essential markers — know which are required.
↑↓
NAND/NOR Forms
Universal-gate equivalents for circuit synthesis.
↘
Step-by-Step
The Quine-McCluskey derivation explained.

Boolean Algebra Laws — Reference

Boolean Algebra Simplifier: Boolean Algebra Laws — Reference
LawOR formAND form
Identity\( A + 0 = A \)\( A \cdot 1 = A \)
Null\( A + 1 = 1 \)\( A \cdot 0 = 0 \)
Idempotent\( A + A = A \)\( A \cdot A = A \)
Complement\( A + \overline{A} = 1 \)\( A \cdot \overline{A} = 0 \)
Double Negation\( \overline{\overline{A}} = A \)
Commutative\( A + B = B + A \)\( A \cdot B = B \cdot A \)
Associative\( (A + B) + C = A + (B + C) \)\( (A \cdot B) \cdot C = A \cdot (B \cdot C) \)
Distributive\( A \cdot (B + C) = AB + AC \)\( A + BC = (A+B)(A+C) \)
Absorption\( A + AB = A \)\( A(A + B) = A \)
De Morgan's\( \overline{A + B} = \overline{A} \cdot \overline{B} \)\( \overline{A \cdot B} = \overline{A} + \overline{B} \)
Consensus\( AB + \overline{A}C + BC = AB + \overline{A}C \)

Input Syntax Cheat Sheet

Boolean Algebra Simplifier: Input Syntax Cheat Sheet
OperatorAccepted FormsExample
AND&, *, ·, ., concatenation AB, word AND, ∧A&B, AB, A AND B
OR+, |, word OR, ∨A+B, A | B, A OR B
NOT~, !, ¬, word NOT, postfix '~A, !A, A', (AB)'
XOR^, ⊕, word XORA XOR B, A^B
NAND⊼, word NANDA NAND B
NOR⊽, word NORA NOR B
Implies->, =>, word IMPLIES, →A -> B
Equivalence<->, <=>, word IFF, ↔A <-> B
Constants0 1 TRUE FALSEA + 0, A * 1

The Quine-McCluskey Algorithm

The Quine-McCluskey method is a systematic, tabular approach to finding the minimal Sum-of-Products expression. Unlike the Karnaugh map (which is visual and limited to about 6 variables), Quine-McCluskey scales and always produces a guaranteed-minimal cover.

  1. List all minterms — each row of the truth table that evaluates to 1.
  2. Group by one-bit count — sort minterms by the number of 1s in their binary representation.
  3. Combine adjacent groups — pairs that differ by exactly one bit merge, replacing that bit with a dash. Repeat until no more combinations are possible.
  4. Collect prime implicants — any term that was never combined further is a prime implicant.
  5. Build the prime-implicant chart — a grid with primes as rows and minterms as columns. Mark which primes cover which minterms.
  6. Select essential primes — any prime that is the only one covering some minterm is essential.
  7. Cover the remaining minterms — choose additional primes to cover minterms not yet covered by essentials, minimizing literal count.

How to Use the Boolean Algebra Simplifier

  1. Enter the expression: Type any Boolean expression using the notation you prefer. You can mix styles — A.B + A'C and A AND B OR NOT A AND C parse identically.
  2. Click Simplify: The tool parses, builds the truth table, runs Quine-McCluskey, and minimizes the expression.
  3. Study the K-map: Each colored loop is a prime implicant. Essential primes are drawn with solid lines, non-essentials with dashes.
  4. Review the steps: The step-by-step panel walks from canonical SOP through prime implicant discovery to the minimal form.
  5. Inspect equivalent forms: Use the minimal POS when your target technology is OR-of-AND, or the NAND-only form when targeting NAND-based silicon.

Applications

Boolean Algebra Simplifier: Applications
FieldUse Case
Digital Circuit DesignMinimize gate count in combinational logic — fewer gates means lower power, smaller die, shorter propagation delay.
Compiler OptimizationSimplify conditional expressions in if-statements and loop guards to reduce branch evaluation cost.
Programmable Logic (FPGA)Fit more logic into available look-up tables by producing minimal-literal forms.
Database Query OptimizationRewrite WHERE-clause predicates into equivalent but cheaper-to-evaluate forms.
Formal VerificationCanonical forms (SOP/POS) enable equivalence checking of two specifications.
Teaching / CourseworkVerify homework, study prime implicant selection, explore K-map grouping strategies.

FAQ

What is Boolean algebra simplification?
Boolean algebra simplification reduces a logic expression to an equivalent expression with fewer operations and literals. A simpler expression means fewer logic gates in hardware, faster evaluation in software, and easier human verification. Methods include algebraic law application, Karnaugh maps, and the Quine-McCluskey algorithm — this tool combines all three.
What is a minimal Sum-of-Products (SOP) form?
The minimal Sum-of-Products form writes a Boolean function as an OR of the fewest-possible product terms, where each product uses the fewest-possible literals. It is found by expanding the function to its canonical SOP (one minterm per true row), then combining adjacent minterms using the rule \( XY + \overline{X}Y = Y \) until no further combination is possible. The result is a minimum two-level AND-OR circuit.
How does a Karnaugh map work?
A Karnaugh map is a grid where rows and columns are labeled with Gray-code bit patterns so that physically adjacent cells differ by exactly one variable. Groups of adjacent 1-cells of sizes 1, 2, 4, or 8 correspond to product terms that eliminate 0, 1, 2, or 3 variables respectively. The largest possible groups give the smallest number of literals per term — so the strategy is "cover every 1 with the largest legal rectangles."
Can I enter minterms and don't-care conditions?
Yes. Type the function as a minterm list such as F(A,B,C,D) = Σm(1,3,7,11,15) + d(0,2,5), or as maxterms with ΠM(…). You can also write m(…) or sum m(…), and dc(…) or x(…) for don't-cares. Don't-cares appear as X in the truth table and K-map and are used only where they make a group larger.
What is a prime implicant?
A prime implicant is a product term that covers a group of minterms and cannot be combined with any other adjacent group to form a larger group. An essential prime implicant is one that is the only prime implicant covering at least one specific minterm — it must appear in every minimal cover. Non-essential primes are optional and only included when needed to cover remaining minterms.
What input syntax does the simplifier accept?
You can write AND as &, *, ·, ., or simple concatenation (AB means A AND B). OR can be + or |. NOT can be ~, !, ¬, or a postfix apostrophe (A', (A+B)'). Word operators AND OR NOT XOR NAND NOR IMPLIES IFF also work, as do arrow forms -> and <->. Parentheses group sub-expressions. Constants 0 and 1 (or TRUE/FALSE) represent fixed values.
Why are NAND-only and NOR-only forms useful?
NAND and NOR are called universal gates because any Boolean function can be built using only one of them. In real silicon, NAND gates are usually the fastest and cheapest to fabricate, so compilers for programmable logic often target a NAND-only netlist. The simplifier shows the equivalent NAND-only and NOR-only expressions — a NAND-only expression can be read directly as a two-level NAND-NAND circuit, which is structurally identical to an AND-OR circuit in switching theory.
How many variables does the tool support?
Up to 8 variables. The Karnaugh map is rendered for 2, 3, and 4 variables (grid sizes 2×2, 2×4, and 4×4 with Gray-code labeling). For 5+ variables the truth table grows to 32+ rows but the Quine-McCluskey algorithm still produces the exact minimal form — the K-map is simply omitted since 3D or split maps are hard to read.
Can I check if two Boolean expressions are equivalent?
Yes — simplify both expressions independently. If their minimal SOP forms are identical, they compute the same function. You can also compare their truth tables row by row. The simplifier outputs both the minterm set and the canonical SOP, which are unique fingerprints of the function.
How do I simplify a logic expression with a truth table output?
Write the function as minterms for the input rows where the output is 1, then enter the minterm indices in the list syntax, such as F(A,B,C) = Σm(1,3,6). The simplifier can use that function to produce a minimal form and its truth table. Check that your variable order matches the row indexing convention you used; changing the order of A, B, and C changes which binary input each index represents.
Why does a simplified Boolean expression have fewer terms but more literals?
Boolean minimization can trade one measure of simplicity for another. An expression with fewer product terms may still contain more variable literals overall, while another equivalent expression may use more terms with shorter products. Several equivalent minimal covers can exist, so compare the form that best fits your circuit or assignment requirements. Also check whether you are comparing SOP with SOP or POS with POS, since those forms group different output values.
What does the apostrophe after a variable mean in a Boolean expression?
A postfix apostrophe means NOT, or the complement of that variable: A' is equivalent to ~A or !A in this simplifier’s accepted notation. For example, AB' means A AND NOT B. A common mistake is placing the apostrophe after a whole expression without parentheses; (A+B)' means NOT of the entire OR expression, while A+B' negates only B. Parentheses make the intended scope clear.

Reference this content, page, or tool as:

"Boolean Algebra Simplifier" at https://MiniWebtool.com/boolean-algebra-simplifier/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-19

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