Math July 13, 2026 · 8 Min Read

Truth Table Generator – Guide & Formulas

Generate truth tables for any Boolean expression. Supports AND, OR, NOT, XOR, NAND, NOR, and implication with step-by-step evaluation.

Generate complete truth tables for any Boolean expression instantly. Enter logical propositions using AND, OR, NOT, XOR, NAND, NOR, and implication operators to see every possible input-output combination.

Key Takeaway

Use the free Truth Table Generator to generate truth tables for any boolean expression. supports and, or, not, xor, nand, nor, and implication with step-by-step evaluation. Get instant results with step-by-step explanations.

How to Use the Truth Table Generator

  1. Enter a Boolean expression using variables (A, B, C) and operators (∧, ∨, ¬, ⊕, →, |, ↓).
  2. Use keyboard shortcuts: * for AND, + for OR, ! for NOT, ^ for XOR, > for implication.
  3. Click "Generate" to produce the full truth table.
  4. Review each row showing all input combinations and the final evaluated output.

The Formula

AND (A ∧ B) is true only when both A and B are true. OR (A ∨ B) is true when at least one operand is true. NOT (¬A) negates the value. XOR (A ⊕ B) is true when exactly one operand is true. NAND (A | B) is ¬(A ∧ B). NOR (A ↓ B) is ¬(A ∨ B). Implication (A → B) is false only when A is true and B is false.

Variable Definitions

  • ∧ (AND): Conjunction — true only when both operands are true
  • ∨ (OR): Disjunction — true when at least one operand is true
  • ¬ (NOT): Negation — inverts the truth value
  • ⊕ (XOR): Exclusive OR — true when exactly one operand is true
  • | (NAND): NOT AND — false only when both operands are true
  • ↓ (NOR): NOT OR — true only when both operands are false
  • → (IMPLIES): Material implication — false only when antecedent is true and consequent is false

Truth Table for (A ∧ B) ∨ ¬C

Generate the truth table for a compound expression with three variables.

  1. Step 1: Identify variables: A, B, C. Total rows = 2³ = 8.
  2. Step 2: List all 8 combinations of A, B, C (000 through 111).
  3. Step 3: Compute A ∧ B for each row (true only when A=1 AND B=1).
  4. Step 4: Compute ¬C for each row (invert C).
  5. Step 5: Compute (A ∧ B) ∨ ¬C by OR-ing the intermediate columns.
  6. Step 6: The output column shows: 1,1,1,0,1,1,0,0 for rows 0–7.

Frequently Asked Questions

What is a truth table?

A truth table is a mathematical table used in logic to compute the output of a Boolean expression for every possible combination of input values. It exhaustively lists all input permutations and their corresponding outputs.

How many rows does a truth table have?

A truth table with n variables has 2ⁿ rows. For example, 2 variables produce 4 rows, 3 variables produce 8 rows, and 4 variables produce 16 rows.

What is the difference between AND and OR?

AND (∧) returns true only when BOTH operands are true. OR (∨) returns true when AT LEAST ONE operand is true. For example, T AND F = F, but T OR F = T.

What is logical implication?

Implication (A → B) is a logical connective that is false only when A is true and B is false. It is equivalent to ¬A ∨ B. In everyday language, it means "if A then B."

Can I use more than 3 variables?

This generator supports expressions with up to 5 variables (A through E), producing truth tables with up to 32 rows. For larger expressions, consider simplifying the formula first.

What is XOR used for?

XOR (exclusive OR) returns true when exactly one operand is true. It is widely used in digital circuits, cryptography, checksums, and toggle logic where two states should be mutually exclusive.

How is NAND different from AND?

NAND (NOT AND) is the negation of AND. It returns false only when both inputs are true, and true otherwise. NAND is a universal gate — any Boolean function can be implemented using only NAND gates.

Can I simplify expressions using truth tables?

Yes. By comparing output columns, you can identify logically equivalent expressions and simplify formulas. Karnaugh maps and Boolean algebra are formal methods that build on truth table analysis.