Boolean Algebra Solver & Truth Table Generator

Truth Table Generator

Karnaugh Map (K-Map) Visualizer

Step-by-Step Simplification

Minimized SOP (from K-Map):
F = ...

Canonical Forms & Lists

Minterms List:
Σm(...)
Maxterms List:
ΠM(...)
Sum of Products (SOP / Canonical Form):
F = ...
Product of Sums (POS / Canonical Form):
F = ...

Enter any Boolean expression using AND, OR, NOT, XOR, NAND, or NOR — and instantly generate its complete truth table with sub-expression steps, Sum of Products (SOP), Product of Sums (POS), minterm/maxterm lists, and a Karnaugh map. Free, no sign-up, works in your browser.

AB or A*B → AND
A+B → OR
!A or A' → NOT
A^B → XOR
NAND(A,B) → NAND
NOR(A,B) → NOR
(A+B)*C → Parentheses
A XNOR B → XNOR

Operator precedence (high to low): NOT → AND → XOR → OR. Use parentheses to override. Variables must be single capital letters A–H.

Boolean algebra logic gate symbols — AND, OR, NOT, NAND, NOR, XOR gates with their truth tables showing input and output combinations
The six standard logic gates with their Boolean symbols and truth tables. Each gate implements one Boolean algebra operation. This solver evaluates any combination of these operators for all possible input values. Image: Wikimedia Commons (CC BY-SA 3.0).

What Is Boolean Algebra?

Boolean algebra is the branch of mathematics that deals with variables that can only take two values: True (1) and False (0). Defined by George Boole in 1854 and extended by Claude Shannon in 1937 to describe electrical circuits, it is the mathematical foundation of every digital system ever built — from the simplest logic gate to a modern CPU with billions of transistors.

In Boolean algebra, variables represent logical conditions (a switch is ON or OFF, a sensor reading is HIGH or LOW) and operators combine them into expressions that describe how a circuit or program behaves. This solver parses your expression and generates a truth table showing every possible input combination and the resulting output — instantly and without manual calculation.

Logic Gates & Boolean Operators

Every Boolean operator corresponds to a physical logic gate in digital electronics. Understanding both the algebraic expression and the gate symbol is essential for digital design coursework.

AND Gate — Conjunction

Output is 1 only when ALL inputs are 1. Equivalent to a series circuit — all switches must be closed for current to flow. Notation: A * B, AB, or A · B.

ABA AND B
000
010
100
111

OR Gate — Disjunction

Output is 1 when ANY input is 1. Equivalent to a parallel circuit — any closed switch allows current to flow. Notation: A + B.

ABA OR B
000
011
101
111

NOT Gate — Inverter

Output is the complement of the input. Turns 1 into 0 and 0 into 1. Every other gate can be built from AND + NOT or OR + NOT combinations. Notation: !A or A'.

ANOT A
01
10

XOR Gate — Exclusive OR

Output is 1 when inputs are different. Used in binary adders, error-detection circuits, and encryption. XOR of a value with itself always returns 0; XOR with 1 always flips the bit. Notation: A ^ B.

ABA XOR B
000
011
101
110

NAND Gate — Universal Gate

NOT AND — output is 0 only when ALL inputs are 1. NAND is a universal gate: any Boolean function can be implemented using only NAND gates. Modern CMOS circuits use NAND gates as the primary building block. Notation: NAND(A,B) = !(A*B).

ABNAND
001
011
101
110

NOR Gate — Universal Gate

NOT OR — output is 1 only when ALL inputs are 0. NOR is also a universal gate. The original TTL logic families (7400-series) used NAND; ECL logic families used NOR as the fundamental cell. Notation: NOR(A,B) = !(A+B).

ABNOR
001
010
100
110

Boolean Algebra Laws & Identities

These laws are used to simplify Boolean expressions into their minimal form — reducing the number of gates needed to implement a circuit. Every step-by-step simplification in this solver names the law applied.

LawAND formOR form
IdentityA · 1 = AA + 0 = A
Null / AnnihilatorA · 0 = 0A + 1 = 1
IdempotentA · A = AA + A = A
ComplementA · A' = 0A + A' = 1
Double Negation!!A = A
CommutativeAB = BAA+B = B+A
Associative(AB)C = A(BC)(A+B)+C = A+(B+C)
DistributiveA(B+C) = AB+ACA+(BC) = (A+B)(A+C)
AbsorptionA(A+B) = AA+AB = A
De Morgan's 1st!(A+B) = !A · !B
De Morgan's 2nd!(A·B) = !A + !B
ConsensusAB + A'C + BC = AB + A'C

Understanding SOP, POS, Minterms & Maxterms

Sum of Products (SOP)

SOP is a canonical form that expresses a Boolean function as an OR of AND terms. For each row where output = 1, write an AND term using all variables (complemented if the variable is 0 in that row). Example: if output is 1 at rows (A=0,B=1) and (A=1,B=1):

F = A'B + AB = B(A' + A) = B

SOP maps directly to a two-level NAND-NAND circuit — the most common implementation in standard cell ASIC design.

Product of Sums (POS)

POS expresses the function as an AND of OR terms. For each row where output = 0, write an OR term using all variables (complemented if the variable is 1 in that row). Example: if output is 0 at row (A=0,B=0):

F = (A + B)

POS maps to a two-level NOR-NOR circuit. SOP and POS always represent the same function — they are dual forms of each other, related by De Morgan's laws.

Minterms (Σm notation)

A minterm is a row where the output equals 1. Each row is numbered by treating the input combination as a binary number. Row A=1, B=0, C=1 is row 5 (101 in binary) = minterm m5. The function written in minterm notation:

F(A,B,C) = Σm(1,3,5,7) — output is 1 at rows 1,3,5,7

Maxterms (ΠM notation)

A maxterm is a row where the output equals 0. Maxterms are the dual of minterms — the same row numbers that are not in the minterm list. The same function in maxterm notation:

F(A,B,C) = ΠM(0,2,4,6) — output is 0 at rows 0,2,4,6

Both representations contain identical information — choose whichever has fewer terms for a more compact expression.

What Is a Karnaugh Map (K-Map)?

4-variable Karnaugh map showing groups of 1s highlighted in different colors for Boolean expression simplification — the groups eliminate variables to produce minimal SOP
A 4-variable Karnaugh map with groups of 1s highlighted. Adjacent cells differ by exactly one variable. Each circled group of 2ⁿ cells eliminates n variables from the expression, producing a simpler SOP term. Larger groups → simpler expressions. Image: Wikimedia Commons (CC BY-SA 3.0).

A Karnaugh Map (K-Map) is a visual method for minimizing Boolean expressions, invented by Maurice Karnaugh in 1953. Instead of applying algebraic laws step by step, you fill in the truth table values on a grid arranged in Gray code order (adjacent cells differ by exactly one variable) and then circle groups of 1s.

The key rules for grouping:

  • Groups must contain 1, 2, 4, 8, or 16 cells (powers of 2)
  • Groups can wrap around the edges of the map (the map is toroidal)
  • Each group must be as large as possible
  • Every 1 must be covered by at least one group
  • Fewer, larger groups → simpler final expression
  • Don't-care conditions (X) can be included in groups if they help make groups larger

A group of 2 cells eliminates 1 variable. A group of 4 eliminates 2 variables. A group of 8 eliminates 3 variables. This is why K-maps produce minimal SOP expressions faster than algebraic simplification for 4-variable functions.

Practical Applications of Boolean Algebra

Digital Circuit Design

Every combinational logic circuit — adders, multiplexers, decoders, comparators, priority encoders — is described by Boolean expressions. Simplifying the expression with K-maps or algebraic reduction directly reduces gate count, silicon area, and power consumption in ASIC and FPGA designs. Use our Digital Logic Simulator to visualize the circuit from any expression this solver generates.

FPGA / VHDL / Verilog Design

Hardware Description Languages (VHDL and Verilog) express logic as Boolean equations. The SOP form output from this solver can be written directly as: assign F = (A & !B) | (!A & B); in Verilog or F <= (A AND NOT B) OR (NOT A AND B); in VHDL. Synthesis tools then minimize and map these to available FPGA LUT resources.

Software Conditional Logic

De Morgan's Laws apply directly to programming conditionals. if (!A && !B) is identical to if (!(A || B)) — the solver proves this by showing both expressions produce identical truth tables. Boolean simplification can reduce nested if chains to simpler, more readable conditions.

Discrete Mathematics & Computer Science

Boolean algebra is the formal foundation of propositional logic taught in discrete mathematics courses. The operators map directly: conjunction (∧) = AND, disjunction (∨) = OR, negation (¬) = NOT, implication (→) = !A + B, biconditional (↔) = XNOR. This solver handles all standard Boolean operators for both digital electronics and discrete math coursework.

Database Query Optimization

SQL WHERE clauses with multiple AND/OR/NOT conditions are Boolean expressions. Query optimizers apply Boolean algebra laws — especially absorption and De Morgan's transformations — to rewrite queries into forms that better match available indexes. Understanding Boolean simplification helps write more efficient queries.

Boolean Search (Google, Legal, Academic)

Advanced search queries use Boolean operators: cat AND dog NOT fish, (python OR javascript) AND tutorial. Understanding precedence (AND before OR, unless parenthesised) is essential for precise results — the same operator precedence rules this solver uses.

Video: Logic Gates, Truth Tables & Boolean Algebra — Complete Tutorial

This comprehensive 54-minute tutorial by The Organic Chemistry Tutor covers everything from binary numbers through AND, OR, NOT, NAND, NOR gates, truth table construction, writing SOP and POS expressions, minterms, maxterms, and Boolean algebra simplification rules. With over 1.3 million views, it is one of the highest-rated introductory digital logic tutorials available — ideal for students taking their first digital electronics or discrete mathematics course.

Logic Gates, Truth Tables, Boolean Algebra AND, OR, NOT, NAND & NOR video tutorial preview

📺 Logic Gates, Truth Tables, Boolean Algebra — AND, OR, NOT, NAND & NOR — The Organic Chemistry Tutor. Covers all six logic gates, truth table construction, SOP/POS forms, minterms, maxterms, and Boolean simplification rules in one complete video.

💡 Seeing "Error 153" or "Video Player Configuration Error"? This occurs when browsers block referrer headers on local offline file views. You can resolve this instantly by clicking here to watch the complete Logic Gates Tutorial directly on YouTube.

Frequently Asked Questions — Boolean Algebra & Truth Tables

Common questions about Boolean algebra, logic gates, truth tables, K-maps, and how to use this solver — answered for students and engineers.

Using the Solver

How do I enter a Boolean expression?

Use capital letters for variables (A, B, C…). AND: AB or A*B. OR: A+B. NOT: !A or A'. XOR: A^B. NAND: NAND(A,B). NOR: NOR(A,B). Use parentheses for grouping: (A+B)*!C. Operator precedence is NOT → AND → XOR → OR.

How many variables can I use?

The solver supports up to 8 variables (A–H), generating a truth table with up to 256 rows. For 4 variables (16 rows), the K-map generator produces a standard 4×4 map. For 2–3 variables, it produces 2×2 and 4×2 maps respectively.

How do I prove two Boolean expressions are equivalent?

Enter both expressions separately and compare their truth tables. If every row produces the same output in both tables, the expressions are logically equivalent. For example, enter !(A+B) and !A*!B — the identical truth tables prove De Morgan's First Law.

What does the sub-expression column feature show?

For complex expressions like (A+B)*(!C), the solver adds intermediate columns showing the result of each sub-expression: first a column for (A+B), then one for (!C), then the final result. This makes it easy to trace how the final output is reached at each row — essential for understanding and debugging logic.

Concepts & Theory

What is SOP vs POS form?

Sum of Products (SOP) is an OR of AND terms — one AND term per row where output = 1. It maps to NAND-NAND circuits. Product of Sums (POS) is an AND of OR terms — one OR term per row where output = 0. It maps to NOR-NOR circuits. Both forms represent the same function and are used depending on which has fewer terms or which gate type is available.

What is a Karnaugh Map and why use it?

A K-Map is a 2D grid of truth table values where adjacent cells differ by exactly one variable. You circle groups of 1s (in powers of 2) to identify terms that can be merged, eliminating variables. A group of 4 eliminates 2 variables from the SOP term. K-maps produce minimal expressions faster than algebraic simplification for functions with 3–4 variables.

What are don't-care conditions in a truth table?

Don't-care conditions (marked X) are input combinations that either never occur or where the output doesn't matter to the design. In K-map simplification, don't-cares can be treated as either 0 or 1 — whichever allows larger groupings and therefore simpler expressions. Common in BCD decoders where inputs 1010–1111 (10–15) are invalid.

Why are NAND and NOR called universal gates?

Because any Boolean function can be built using only NAND gates — or only NOR gates. AND, OR, and NOT are each constructable from 2–3 NAND gates. This is why CMOS standard cells in modern chips are almost entirely NAND and NOR based — a single gate type simplifies fabrication, layout, and timing analysis. Use our Digital Logic Simulator to verify any NAND-only equivalence circuit interactively.

What is the difference between XOR and XNOR?

XOR outputs 1 when inputs are different: A^B = A'B + AB'. XNOR outputs 1 when inputs are the same: A XNOR B = AB + A'B' — it is the complement of XOR. XNOR is used in equality comparators, error detection circuits, and CRC generators. XOR is used in binary adders and cryptographic operations.

What is the Consensus Theorem?

The Consensus Theorem states: AB + A'C + BC = AB + A'C. The term BC is redundant and can be eliminated because it is "covered" by the other two terms. This is a less obvious simplification that algebraic manipulation and K-maps can both identify, but which students often miss when simplifying by hand.

Advanced Boolean Concepts

Duality Principle

Every Boolean theorem has a dual: swap AND↔OR and 0↔1 throughout and the resulting expression is also true. Example: Identity law A + 0 = A has dual A · 1 = A. De Morgan's laws are duals of each other. This means every circuit can be built in two equivalent forms — one using AND/OR and one using the complementary operators.

Functional Completeness

A set of gates is functionally complete if it can implement any Boolean function. Minimum complete sets include: {NAND}, {NOR}, {AND, NOT}, {OR, NOT}. The set {AND, OR} alone is NOT complete (cannot implement NOT). {XOR, AND} is also not complete. This is why every digital IC library includes NAND and NOR as its primary cells — they provide functional completeness with optimal CMOS transistor efficiency (4 transistors per 2-input gate).

Shannon's Expansion Theorem

Any Boolean function can be expanded around a variable: F(A,B,C) = A · F(1,B,C) + A' · F(0,B,C). This is the theoretical basis of how FPGA lookup tables (LUTs) work — a 4-input LUT stores all 16 values of a 4-variable truth table in a small SRAM, implementing any 4-variable Boolean function without re-routing. Shannon's expansion also underlies binary decision diagrams (BDDs) used in formal verification.

Quine-McCluskey Method

The Quine-McCluskey algorithm is a tabular method for Boolean minimization that works for any number of variables — unlike K-maps which become impractical above 5–6 variables. It systematically finds all prime implicants (the largest possible groupings) and selects a minimal cover. This is the algorithm used by logic synthesis tools in EDA software like Yosys, Synopsys Design Compiler, and Cadence Genus — the same tools used to synthesize Verilog and VHDL for ASIC and FPGA production chips.

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