Truth Table Generator

Frequently Asked Questions

What does a truth table show?

Every possible combination of true/false inputs and the resulting value of the whole Boolean expression.

Which operators are supported?

AND, OR, NOT, XOR, and implication (→), with parentheses for grouping and any number of variables.

How many rows will there be?

2ⁿ rows for n distinct variables - 2 variables give 4 rows, 3 give 8, 4 give 16.

How do I spot a tautology?

If the result column is true in every row the expression is a tautology; all false is a contradiction; a mix is contingent.

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How This Calculator Works

A truth table lists every possible combination of true/false inputs for a Boolean expression and the resulting output. With n variables there are 2ⁿ rows - each variable independently true or false. This tool ships a hand-written recursive-descent parser (no JavaScript eval()) that accepts both words and symbols: NOT / !, AND / &, OR / |, XOR / ^, IMPLIES / −>, and IFF / <−>. Precedence runs ¬ (tightest), then ∧, ⊕, ∨, →, and ↔ (loosest); parentheses override. It evaluates the expression on all 2ⁿ input rows and then classifies it: a tautology is true in every row, a contradiction false in every row, and a contingency is true for some rows and false for others.

Worked Example: A ∧ B → C

Three variables give 2³ = 8 rows. Implication P → Q is false only when P is true and Q is false. Here the antecedent A ∧ B is true only when both A and B are true; in that single situation the whole formula collapses to C. So the expression is false in exactly the two rows where A = B = true and C = false - it is a contingency. Contrast A ∨ ¬A, which is true in every row (a tautology - the law of excluded middle), and A ∧ ¬A, false in every row (a contradiction).

Where Truth Tables Are Used

Digital logic: every combinational circuit (adders, multiplexers) is specified by a truth table before gate synthesis.

Programming: verifying that complex if-conditions and guard clauses cover every case without dead branches.

Proofs: a propositional argument is valid exactly when the implication premises → conclusion is a tautology.

Hardware verification & SAT solvers: satisfiability is "does any row come out true?"

Common Mistakes & Edge Cases

Implication is not symmetric: A → B ≠ B → A, and A → B is true whenever A is false (vacuous truth) - a frequent surprise.

XOR (⊕) is true when inputs differ; IFF (↔) is its negation, true when they match.

Precedence matters: A ∨ B ∧ C means A ∨ (B ∧ C); use parentheses when unsure.

Variables are A–E only and case-insensitive; more than five would need 64+ rows and is rejected.

De Morgan: ¬(A ∧ B) ≡ ¬A ∨ ¬B - a classic spot where dropping a negation flips the whole table.

Reading the Five Connectives

It helps to memorise each connective by its "false" (or "true") row, since that is usually the rare case. AND (∧) is true only when every input is true. OR (∨) is the inclusive or - false only when all inputs are false. XOR (⊕) is true when an odd number of inputs are true, which for two variables means "exactly one". IMPLIES (→) is false in only one of its four rows: true antecedent with false consequent; in particular "false implies anything" is vacuously true, the rule that most often trips students. IFF (↔, biconditional) is true exactly when both sides agree, so it is the negation of XOR. Two formulas are logically equivalent precisely when their result columns match in every row - that is how you prove identities like the distributive law or De Morgan's by table rather than by algebra.

Related Tools & Limits

Truth tables connect to Karnaugh maps and Boolean algebra simplification (minimising the gate count of a circuit), and to propositional logic proofs. They are complete and decisive but grow exponentially: 10 variables already mean 1,024 rows, and 20 variables over a million, so for large formulas SAT solvers, binary decision diagrams, and Quine−McCluskey minimisation replace exhaustive enumeration. This tool covers propositional logic only - it has no quantifiers (∀, ∃), which belong to predicate (first-order) logic and cannot be settled by any finite table.

Frequently Asked Questions

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