How do you simplify a boolean expression?

Use boolean algebra laws: (1) Complement: A·A' = 0, A+A' = 1; (2) Absorption: A+A·B = A; (3) De Morgan's: (A·B)' = A'+B'; (4) Consensus: A·B + A'·C + B·C = A·B + A'·C. For systematic minimization, use the Quine-McCluskey algorithm or Karnaugh maps.

What is a truth table in boolean algebra?

A truth table lists all possible combinations of input variables (0s and 1s) and the corresponding output for each combination. For n variables there are 2^n rows. For example, 2 variables A and B have 4 rows: 00, 01, 10, 11.

What is SOP (Sum of Products) form?

SOP is a canonical form where the boolean function is expressed as an OR (sum) of AND (product) terms. Each AND term is a minterm — a row where the output is 1. Example: F = A'B + AB' is in SOP form. The canonical (full) SOP includes all variables in each term.

What is a Karnaugh map (K-map)?

A Karnaugh map is a visual method for simplifying boolean expressions. Variables are arranged in a grid using Gray code ordering (00, 01, 11, 10) so adjacent cells differ by exactly one variable. Group 1s in powers of 2 (1, 2, 4, 8) to eliminate variables and find the minimal SOP.

What is De Morgan's theorem?

De Morgan's laws state: (1) (A·B)' = A' + B' — the complement of an AND is an OR of complements; (2) (A+B)' = A'·B' — the complement of an OR is an AND of complements. These are used to convert between NAND/NOR expressions and AND/OR expressions.

What are minterms and maxterms?

A minterm is a product term where the function equals 1. A maxterm is a sum term where the function equals 0. For n variables, there are 2^n possible minterms (m0 to m_{2^n-1}). A boolean function can be expressed as a sum of minterms (SOP) or product of maxterms (POS).

How does the Quine-McCluskey algorithm work?

Quine-McCluskey systematically minimizes boolean functions: (1) List all minterms as binary numbers; (2) Group by number of 1-bits; (3) Combine adjacent groups (differ by 1 bit) and mark with a dash; (4) Repeat until no more combinations; (5) Uncombined terms are prime implicants; (6) Select minimum set of essential prime implicants to cover all minterms.

Boolean Algebra Calculator

Our Boolean Algebra Calculator handles all digital logic and boolean algebra operations in one place. Enter any boolean expression using standard operators — AND, OR, NOT, XOR, NAND, NOR, XNOR — and instantly get the truth table, minterms, maxterms, canonical SOP and POS forms, and a minimized expression using the Quine-McCluskey algorithm. The Karnaugh map visualizer highlights which cells are minterms, making it easy to identify groupings and simplifications.

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calendar_today Updated: July 2025

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Boolean Expression

Operators: * or AND = AND | + or OR = OR | ' or ! = NOT | ^ or XOR = XOR | NAND, NOR, XNOR

Examples:

Minimal SOP (Simplified)

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Minterms Σm(…)

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Maxterms ΠM(…)

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Canonical SOP

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Canonical POS

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Truth Table

calculate Boolean Laws

Identity

A+0=A A·1=A

Null

A+1=1 A·0=0

Idempotent

A+A=A A·A=A

Complement

A+A'=1 A·A'=0

Absorption

A+AB=A A(A+B)=A

De Morgan's

(AB)'=A'+B'

(A+B)'=A'·B'

Double Negation

(A')'=A

grid_on Input Syntax

AND A*B A·B A AND B

OR A+B A OR B

NOT A' !A NOT A

XOR A^B A XOR B

NAND A NAND B

NOR A NOR B

lightbulb Quick Tips

•AB means A AND B (implicit multiplication)
•K-map groups must be powers of 2: 1, 2, 4, 8
•Green cells = 1 (minterm), Red = 0 (maxterm)
•NAND and NOR are universal gates
•More 1s grouped → simpler expression

How to Use This Calculator

Enter Expression

Type your boolean expression using * (AND), + (OR), ' or ! (NOT), ^ (XOR). Variables are single letters A-Z.

View Truth Table

The truth table is generated automatically for all variable combinations (up to 4 variables = 16 rows)

Check K-map

The Karnaugh map highlights minterms (1s) making it easy to spot groupings for simplification

Get Simplified Form

See the minimal SOP expression from the Quine-McCluskey algorithm along with canonical SOP/POS forms

The Formula

A boolean function is fully defined by its truth table. SOP canonical form: OR of all minterms (rows where output = 1). POS canonical form: AND of all maxterms (rows where output = 0). The Quine-McCluskey algorithm minimizes SOP by grouping minterms that differ by one variable.

F = Σm(minterms) = Πm(maxterms)

lightbulb Variables Explained

A, B, C, D Boolean input variables (0 or 1)
AND (·) Logical AND — output 1 only if both inputs are 1
OR (+) Logical OR — output 1 if at least one input is 1
NOT (') Logical NOT — inverts the input
XOR (⊕) Exclusive OR — output 1 if inputs differ
SOP Sum of Products — OR of AND terms (minterms)
POS Product of Sums — AND of OR terms (maxterms)

tips_and_updates Pro Tips

Use * or · for AND, + for OR, ' or ! for NOT, ^ for XOR

Complement law: A + A' = 1, A · A' = 0 — use this to eliminate variables

De Morgan's: (A·B)' = A'+B' and (A+B)' = A'·B'

Absorption: A + A·B = A and A·(A+B) = A

Group minterms in powers of 2 (1, 2, 4, 8) on the Karnaugh map for minimal SOP

NAND and NOR are universal gates — any boolean function can be built from just NAND

Simplify Boolean Expressions and Generate Truth Tables Instantly

Boolean algebra is the mathematical foundation of digital electronics, computer science, and logic circuit design. Every processor, memory chip, and programmable logic device relies on boolean operations — AND, OR, NOT, XOR, NAND, NOR — to process binary data. Engineers and computer science students routinely need to simplify boolean expressions to minimize gate counts, reduce circuit complexity, and optimize power consumption. A single 4-variable function can have up to 16 rows in its truth table and dozens of possible simplifications. Manual simplification using De Morgan's theorems, consensus, and absorption laws is tedious and error-prone, especially under exam conditions. This boolean algebra calculator automates the entire workflow: enter any expression with up to four variables, and it instantly produces the complete truth table, identifies all minterms and maxterms, generates canonical Sum-of-Products (SOP) and Product-of-Sums (POS) forms, applies the Quine-McCluskey algorithm for guaranteed minimal expressions, and renders a color-coded Karnaugh map for visual verification. Whether you are designing combinational logic for an FPGA, verifying homework problems, or preparing for a digital logic exam, this tool replaces hours of manual work with accurate, step-by-step results.

What is a Boolean Algebra Calculator?

A boolean algebra calculator evaluates and simplifies logical expressions using the rules of boolean algebra. It generates truth tables showing every input/output combination, identifies minterms and maxterms, and applies minimization algorithms like Quine-McCluskey to find the simplest equivalent expression.

Boolean Operators and Notation

This calculator accepts multiple notation styles: AND (*, ·, AND, &&), OR (+, OR, ||), NOT (', !, ~, NOT), XOR (^, XOR), NAND, NOR, XNOR. Variables are single letters A through D. Parentheses are supported for grouping. Examples: A*B+C', (A OR B) AND NOT C, A NAND B.

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