How is a truth table read and interpreted?

A truth table lists every possible combination of inputs (each variable = true or false) and shows the output for each combination. For n variables, there are 2ⁿ rows. The rows are typically ordered in binary counting order: F,F,F = 0,0,0 → row 1; F,F,T = 0,0,1 → row 2; ...; T,T,T = 1,1,1 → row 8. An expression is a tautology if every row is true, a contradiction if every row is false, and satisfiable if at least one row is true. Truth tables exhaustively verify whether two expressions are logically equivalent — if their output columns match row for row, the expressions are equivalent.

What is the difference between AND (∧) and NAND?

AND (∧) returns true only when both inputs are true: 0∧0=0, 0∧1=0, 1∧0=0, 1∧1=1. NAND (NOT-AND) is simply the negation of AND: it returns false only when both inputs are true — 0 NAND 0 = 1, 0 NAND 1 = 1, 1 NAND 0 = 1, 1 NAND 1 = 0. NAND is notable for being functionally complete — any Boolean function can be built from NAND gates alone. Similarly, NOR (NOT-OR) returns true only when both inputs are false.

Can Boolean algebra simplify my code's if-statements?

Absolutely. Many complex conditional chains can be reduced using Boolean identities. A common pattern: `if ((x && y) || (!x && z))` is equivalent to `if (x ? y : z)` — the ternary captures the same logic. Another: `if (!(a && b))` simplifies to `if (!a || !b)` by De Morgan. Simplifying conditions reduces cognitive load, improves readability, and eliminates subtle bugs from unintended interactions between nested conditions.

How many variables can the Boolean calculator handle?

Our calculator supports up to 8 variables, which produces a 256-row truth table. Beyond 8, the truth table becomes unwieldy and algebraic simplification is the more practical approach. For expressions with many variables, the calculator switches to algebraic mode and uses consensus theorem, absorption, and distribution rules to simplify without enumerating every combination.

What are minterms and maxterms in Boolean algebra?

A minterm is an AND term where every variable appears exactly once (in true or complemented form) — it corresponds to exactly one row of the truth table where the output is 1. A maxterm is an OR term where every variable appears exactly once — it corresponds to exactly one row where the output is 0. Any Boolean function can be expressed as a sum of minterms (canonical SOP / Sum of Products) or a product of maxterms (canonical POS / Product of Sums). These canonical forms are unique for each function, meaning two functions are equivalent if and only if they have the same canonical form.

Calculators2026-05-10

Boolean Algebra Calculator: Truth Tables, De Morgan's Laws, and Logic Simplification

Every "if" statement you write, every database query you filter, and every search engine result you see is powered by Boolean logic — a mathematical system where every variable is strictly true or false (1 or 0). George Boole published this algebra in 1854 in "The Laws of Thought," and it sat quietly in mathematics departments for nearly a century before Claude Shannon realised it could describe electrical switching circuits, laying the foundation for every digital computer ever built. Today, Boolean algebra underpins everything from SQL WHERE clauses to search engine queries to the design of microprocessors with billions of logic gates. Our free Boolean algebra calculator lets you enter any Boolean expression (up to 8 variables), instantly see its truth table, simplify it using algebraic rules and De Morgan's laws, and view the equivalent logic gate diagram — all running in your browser.

rule

Boolean Logic Calculator

Free · No registration

Try this tool for free →open_in_new

Step-by-Step Guide

Enter Your Boolean Expression

Type your expression using standard notation: & or ∧ for AND, | or ∨ for OR, ! or ¬ for NOT, ^ or ⊕ for XOR. You can use parentheses for grouping, and variable names like A, B, C, up to H. Example: (A & B) | (!A & C) represents "either A and B are both true, or A is false and C is true." The calculator validates syntax in real time and shows you an auto-formatted version of your expression with proper mathematical symbols.

View the Truth Table

The truth table displays every possible combination of input values (2ⁿ rows for n variables) and the resulting output. Each row shows whether the expression evaluates to true or false given that specific input. For 3 variables (A, B, C), you'll see 8 rows; for 4 variables, 16 rows. The truth table is colour-coded — green for true outputs, grey for false — so you can instantly scan for patterns. This is the fastest way to understand what a Boolean expression actually means.

Apply Simplification and See the Circuit

Click "Simplify" and the calculator applies Boolean algebra identities to reduce your expression to its simplest form. The step-by-step panel shows which rule was applied at each stage: idempotent law (A&A=A), complement law (A&!A=0), absorption law (A|(A&B)=A), and De Morgan's laws. The simplified expression is shown alongside a logic gate diagram using standard ANSI/IEEE symbols — AND (flat-back), OR (curved-back), NOT (triangle with bubble), XOR (OR with extra line).

Tips & Best Practices

check_circle

De Morgan's laws are the two most powerful Boolean identities: !(A & B) = !A | !B, and !(A | B) = !A & !B. They let you convert ANDs to ORs and vice versa while pushing NOT inward. In practical terms: "not (raining AND cold)" equals "not raining OR not cold." These laws are essential for simplifying complex SQL WHERE clauses and search engine queries.

check_circle

The XOR (exclusive OR) operator returns true when exactly one input is true — it's the "one or the other, but not both" operator. XOR with a constant 1 is the same as NOT: A ⊕ 1 = !A. XOR is also its own inverse: (A ⊕ B) ⊕ B = A. This self-inverse property makes XOR the foundation of many encryption schemes and basic checksum algorithms.

check_circle

NAND and NOR are called "universal gates" because you can build AND, OR, and NOT using only NAND gates (or only NOR gates). Every digital circuit in existence can theoretically be reduced to a network of NAND gates. In fact, NAND flash memory (used in SSDs and USB drives) gets its name from this very gate.

check_circle

When writing Boolean conditions in code, the order of conditions matters for short-circuit evaluation. In `if (cheapCheck() && expensiveCheck())`, if cheapCheck() returns false, expensiveCheck() never runs. Put your fastest-failing condition first — it's a simple optimisation that compounds in tight loops.

check_circle

Truth tables grow exponentially: n variables means 2ⁿ rows. At 8 variables, that's 256 rows — still manageable. At 32 variables, you'd need over 4 billion rows. This is why formal verification of large digital circuits uses SAT solvers and Binary Decision Diagrams rather than brute-force truth tables.

check_circle

SQL's three-valued logic (TRUE, FALSE, NULL) extends classic Boolean algebra. In SQL: NULL AND FALSE = FALSE (short-circuit), but NULL AND TRUE = NULL (unknown), and NULL OR TRUE = TRUE (short-circuit). When writing database queries with nullable columns, always account for NULL propagation — it's one of the most common sources of subtle query bugs.

check_circle

Karnaugh maps (K-maps) are a visual alternative to algebraic simplification for up to 4 variables. Arrange the truth table in a grid where adjacent cells differ by exactly one variable; circle groups of 1s in sizes of 1, 2, 4, or 8; each group corresponds to a simplified product term. Our calculator shows the K-map for expressions with 2-4 variables alongside the algebraic simplification.

Frequently Asked Questions

De Morgan's laws state: (1) NOT (A AND B) = (NOT A) OR (NOT B), and (2) NOT (A OR B) = (NOT A) AND (NOT B). They describe how negation distributes over AND/OR. Practical applications: negating a compound SQL condition — `WHERE NOT (status='active' AND age>18)` becomes `WHERE status!='active' OR age<=18`; writing negative search queries — "dogs NOT (poodles OR chihuahuas)" becomes "dogs NOT poodles AND NOT chihuahuas"; simplifying circuit designs by pushing inverters through gates.

Boolean algebra is the hidden grammar of computing — from the logic gates in your CPU to the WHERE clause in your SQL to the search operators in Google. Our calculator makes this algebra visible, simplifying expressions, generating truth tables, and drawing the circuits your logic describes. Try it now and see what your conditions really mean.

Try this tool for free →open_in_new