Truth table generator for any logic expression

This online tool builds the truth table of a logic expression with an intermediate column for every operation, in the order they are evaluated. It reads mathematical notation (¬ ∧ ∨ → ≡), programming (! && ||), engineering (A'B + AB') and words (AND, OR, NOT), shows how the expression was parsed, and derives the full and minimal DNF and CNF, a Karnaugh map, the Zhegalkin polynomial and a logic gate diagram from the table.

How to build a truth table

1

Type the expression — for example, (A → B) ∧ ¬C — or tap an example. The symbol keys below the field help on a phone.

2

The table appears right away, with a numbered column for each operation. Check the "Read as" line to see how brackets were placed.

3

Switch tabs for the normal forms, the Karnaugh map, the Zhegalkin polynomial or the circuit; copy the table into Excel or save it as PDF.

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Build the table, simplify the expression and read off every normal form

Logic expression

Examples:
Reading and ordering

Column order can also be fixed in the expression itself: F(z, y, x) = …. A variable listed there but absent from the expression becomes a dummy one.

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Features

A column for every operation
Intermediate columns are numbered in evaluation order — brackets first, then NOT, AND, OR, implication, equivalence — the way it is done by hand.
Any notation at once
¬ ! ~ not, ∧ & && *, ∨ | + or, → -> =>, ≡ <-> ==, ⊕ ^ xor, NAND ↑ and NOR ↓, the prime A' and AB written together — and a note wherever a sign could be read two ways.
Full and minimal forms
Full DNF and CNF with the rows each term comes from, minimal DNF and CNF by Quine–McCluskey with the merging stages and the prime implicant chart, and a step-by-step simplification naming each law.
Clickable Karnaugh map
For 2 to 6 variables in Gray code, with colored groups linked to the terms; clicking a cell changes the function and the formulas follow.
Column puzzle solver
Given an expression and a fragment of its table with unlabeled columns, it finds which column is which variable, fills the forced cells and explains the reasoning.
For discrete math and circuits
Zhegalkin polynomial by the triangle method, Post's classes with a counterexample for each, dummy variables, and a gate diagram in ANSI, IEC or GOST symbols as PNG or SVG.

What you can enter

  • A school expression: (x → y) ∧ ¬(y ≡ z)
  • An engineering one: A'B + AB'C, (A + B)(A' + C)
  • A programming one: !a && (b || c)
  • In words: not a and (b or c)
  • Several expressions separated by a semicolon to check equivalence: A → B; ¬A ∨ B
  • A value string instead of a formula: 01101001, or F(a, b, c) = 0110 1001

Signs and priorities

NOT: ¬ ! ~ not, a prime A' after an operand. AND: ∧ & && * · and, or letters written together. OR: ∨ | || + or. Exclusive OR: ⊕ ^ xor !=. Implication: → -> => ⊃ implies, and <= as in Python solutions; reverse implication ← <- >=. Equivalence: ≡ ↔ <-> <=> == =, and ~ between operands. NAND: ↑ nand; NOR: ↓ nor. Constants: 0 1 true false. Priority: NOT, then AND, then OR and XOR, then implication, then equivalence; brackets ( ) [ ] { } change the order.

How this differs from a plain truth table generator

A plain generator prints the final column and stops. This one shows the working: every operation gets its own numbered column, so a filled-in table can be checked cell by cell, and a practice mode lets you fill the columns yourself and see which cells are wrong. The second difference is that it says how your expression was read — a chain of implications, the sign ~ or the letters AB can be read two ways, and the tool names its choice and offers the other one. The third is everything that follows from the table: forms with their origin, minimization with its steps, a Karnaugh map you can click to design a function, and a circuit.

Frequently asked questions

In what order are the logical operations evaluated?

Brackets first, then negation, then conjunction (AND), then disjunction (OR) together with exclusive OR, then implication, and equivalence last. So ¬A ∧ B ∨ C ∧ D means ((¬A) ∧ B) ∨ (C ∧ D). One case has no single rule: a chain of implications without brackets. Mathematical logic reads A → B → C as A → (B → C), while some school courses evaluate it left to right. Implication is not associative, so the two readings give different tables — the tool shows which one it used and lets you switch.

Why is an implication true when the premise is false?

A → B is a promise: "if A, then B". It is broken only when A holds and B does not — that is the single row 1 → 0 = 0. When A is false the promise was never tested, so it counts as kept, and the implication is 1. That is why A → B has the same table as ¬A ∨ B, and why the tool replaces implication with exactly this formula when simplifying.

Why is the full DNF so long when the minimal DNF is short?

The full DNF takes one conjunction of all variables for every row where F = 1, so a function that is 1 on six rows gets six long terms. Minimization merges terms that differ in one variable: A∧B∧C ∨ A∧B∧¬C = A∧B. On a Karnaugh map such terms are neighboring cells, and a group of 2, 4 or 8 cells becomes one short term. The Quine–McCluskey method does the same merging systematically and then picks the cheapest set of terms that still covers every 1.

How do you find which unlabeled column is which variable?

List the rows of the full table where F takes the value given in the fragment — there are usually few of them. Look for what they share: if y = 1 whenever F = 0, a column with a 0 in an F = 0 row cannot be y. Then take the most informative row of the fragment and see which full rows it can be; the rows of the fragment are all different, which rules out more. The column puzzle mode does exactly this and writes out every step.

What are the Zhegalkin polynomial and Post's classes for?

The Zhegalkin polynomial writes any Boolean function with only AND, XOR and the constant 1, and it is unique for each function — a convenient way to compare functions and to test linearity. Post's five classes answer another question: can every function be built from a given set of gates? By Post's theorem a set is complete exactly when, for each of the five classes, it contains a function outside that class. NAND and NOR lie in none of them, which is why each of them alone is enough to build any circuit.

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