Logic Gate Truth Table Generator

Generate complete truth tables for any Boolean expression. Supports AND, OR, NOT, XOR, and IMPLIES (→) operators with up to 6 variables. Instant results, interactive & educational.

Variables will be automatically detected from your expression
Client‑side computation: Your expressions never leave your browser. Fast, secure, and private.
Verification: (p → q) ∧ q

This verification table shows the truth values for the expression: (p → q) ∧ q

p q (p → q) ∧ q
F F F
F T T
T F F
T T T

Analysis: The expression is true in 2 out of 4 cases (50.0%). This expression is contingent (true in some cases, false in others).

Understanding Truth Tables & Boolean Logic

A truth table is a mathematical table used in logic—specifically in Boolean algebra and digital circuit design—to list all possible combinations of truth values (True/False or 1/0) for input variables, alongside the resulting output of a logical expression. Truth tables form the foundation of propositional calculus, computer architecture, and algorithmic reasoning.

For any Boolean function f(x₁, x₂, ..., xₙ), the truth table exhaustively maps 2ⁿ possible input states to the output.

Operators supported:
AND (∧) : true only if both operands true
OR (∨) : true if at least one operand true
NOT (¬) : inverts truth value
XOR (⊕) : true if operands differ
IMPLIES (→) : false only when antecedent true and consequent false (p→q ≡ ¬p ∨ q)
IFF (↔) : true if both operands have same truth value (biconditional)

Why Use a Truth Table Generator?

  • Digital Logic Design: Validate logic gates, combinational circuits, and arithmetic units (adders, multiplexers).
  • Computer Science Education: Learn Boolean identities, simplify expressions, and test logical equivalence.
  • Software Engineering: Debug complex conditionals, verify state transitions, and reason about code paths.
  • Mathematics & Philosophy: Analyze propositional statements, tautologies, contradictions, and contingencies.

How the Generator Works (Step‑by‑Step)

  1. Parse variables: Extract variable names from your comma‑separated list (e.g., p, q, r). Case‑insensitive, single letters only.
  2. Generate all combinations: For n variables, the tool creates 2ⁿ rows (from all false to all true).
  3. Normalize expression: Translate logical keywords (AND, OR, NOT, XOR, IMPLIES, IFF) and symbols (→, ->, ↔, <->) to JavaScript operators (&&, ||, !, ^, and p→q becomes !p || q, p↔q becomes p === q).
  4. Safely evaluate: For each row, substitute variable values into the transformed expression and compute the Boolean result using a sandboxed Function — no eval risk.
  5. Render table: Display the final truth table with column headers and row highlighting.

Real‑world Engineering Applications

? Digital Circuit Design: 2‑to‑1 Multiplexer (MUX)

A 2‑to‑1 multiplexer selects between two inputs (A, B) based on a control signal (S). Its Boolean expression: Output = (¬S ∧ A) ∨ (S ∧ B). Using this tool with variables (S, A, B), you can verify all 8 scenarios, confirming that when S=0, Output=A, and when S=1, Output=B.

? Software Condition Testing: Complex Conditional Logic

Validate complex conditionals in programming: if ((user.isAdmin || user.isModerator) && (post.isPublic || overrideFlag)) { ... }. Model as (A ∨ B) ∧ (C ∨ D) with 4 variables, generating 16 rows to ensure all access‑control cases are handled correctly.

? Traffic Light Controller Logic

Consider a simplified intersection controller with sensors: CarA (vehicle on road A), CarB (vehicle on road B), and a Timer override. The green light for road A activates if CarA is present AND (CarB is absent OR Timer is active): GreenA = CarA ∧ (¬CarB ∨ Timer). Generate the truth table to verify all 8 scenarios.

Advanced Topic: Logical Completeness & Expressive Power

You might wonder: why only these operators? The sets {NOT, AND} or {NOT, OR} are each functionally complete—they can express any Boolean function. Our tool provides {NOT, AND, OR, XOR, IMPLIES, IFF} for intuitive expression.

Use this tool to verify equivalences like: p XOR q ≡ (p ∨ q) ∧ ¬(p ∧ q) and p → q ≡ ¬p ∨ q.

Learning Tip: Start with the "Hypothetical Syllogism" example (p→q ∧ q→r) to observe how intermediate variable q cancels out, demonstrating a classic logical deduction pattern.

Boolean Laws & Simplification

  • Identity: A ∧ 1 = A, A ∨ 0 = A
  • Null: A ∧ 0 = 0, A ∨ 1 = 1
  • Idempotent: A ∧ A = A, A ∨ A = A
  • Complement: A ∧ ¬A = 0, A ∨ ¬A = 1
  • De Morgan’s: ¬(A ∧ B) = ¬A ∨ ¬B
  • Distributive: A ∧ (B ∨ C) = (A∧B) ∨ (A∧C)
  • Absorption: A ∨ (A ∧ B) = A
  • Implication: A → B ≡ ¬A ∨ B

Common Logical Equivalences (Verified by Truth Tables)

EquivalenceDescription
p → q ≡ ¬p ∨ qImplication to OR
p ↔ q ≡ (p → q) ∧ (q → p)Biconditional definition
¬(p ∧ q) ≡ ¬p ∨ ¬qDe Morgan's Law (AND)
¬(p ∨ q) ≡ ¬p ∧ ¬qDe Morgan's Law (OR)
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)Distributive (OR over AND)
p XOR q ≡ (p ∨ q) ∧ ¬(p ∧ q)XOR definition
p ↔ q ≡ (p ∧ q) ∨ (¬p ∧ ¬q)Biconditional equivalence

Complexity note: Generating a truth table for n variables requires O(2ⁿ) computations. For n=6, that's 64 rows; for n=10, 1024 rows. Our tool limits to 6 variables for real‑time responsiveness. For complex expressions, use Karnaugh Maps for simplification.

Frequently Asked Questions

You can use uppercase or lowercase variables (A–Z or a–z), operators AND/OR/NOT/XOR/IMPLIES/IFF, symbols && || ! ^ -> → ↔ <->, parentheses ( ), and spaces. Implication can be written as "IMPLIES", "->", or "→". Biconditional as "IFF", "↔", or "<->".

For performance and clarity, we support up to 6 variables (2⁶ = 64 rows). More than 6 may cause slow rendering; the tool will warn you.

Absolutely. The generator sanitizes inputs, restricts to allowed characters, and uses a controlled Function constructor without direct eval. No external code execution.

Yes. If the output column is all True, the expression is a tautology. If all False, it's a contradiction. The table instantly reveals logical properties.

Generate truth tables for both expressions separately and compare the output columns. If they match for every row (all 2ⁿ combinations), the expressions are equivalent. Alternatively, build a single expression: expr1 ↔ expr2 (biconditional). If the result is a tautology (all True), they are equivalent.
Academic references: Wolfram MathWorld – Truth Table, Shannon, C. E. (1938). "A Symbolic Analysis of Relay and Switching Circuits", Boolean Algebra (Wikipedia).
Reviewed by GetZenQuery Tech team — updated April 2026. Implication operator implemented following standard material conditional semantics.