Truth Table Generator (AND, OR, NOT, XOR, implies, iff)

Understanding the truth table generator — concept and principle

This tool builds the truth table of a logical expression and tells you whether it is a tautology, a contradiction or neither. It lists every row, the result column, the number of true rows and which rows are true. Use it for logic and discrete mathematics courses, digital circuit design, programming conditions and checking arguments.

How it is calculated

• With n variables there are 2ⁿ rows, from all true (T) to all false (F), with the first variable changing slowest.
• ¬p is true when p is false. p ∧ q is true only when both are true. p ∨ q is true when at least one is true. p ⊕ q is true when exactly one is true.
• p → q is false only when p is true and q is false. p ↔ q is true when both have the same value.
• Binding order: ¬ first, then ∧, then ∨ and ⊕, then →, then ↔; → groups to the right. Use brackets to be explicit.

A statement true in every row is a tautology; false in every row, a contradiction.

The principle

A compound statement's truth depends only on the truth of its parts. Listing every combination therefore covers every possible situation, and the multiplication principle says there are 2 × 2 × … = 2ⁿ combinations. Each row is evaluated from the inside out, like arithmetic. A tautology is a statement that is true by its logical form alone, whatever the facts. Two expressions are logically equivalent when their result columns match in every row.

Worked example

Default expression: (p → q) ∧ (q → r) → (p → r).

1. Three variables give 2³ = 8 rows.
2. Row T T T: p → q = T, q → r = T, p → r = T, so the whole is T.
3. Row T F T: p → q = F, so the left side is F, and F → anything is T.
4. Row T T F: q → r = F, the left side is F, so the result is T.
5. Every row is true: TTTTTTTT, 8 of 8. Tautology — always true. This is the hypothetical syllogism (transitivity of implication).

How to read the result

A tautology is a valid rule of reasoning. A contradiction can never be true. Anything else is contingent: the true rows tell you exactly in which situations it holds. To check whether two expressions are equivalent, test A ↔ B; if it is a tautology, they are equivalent.

Common mistakes

• Treating p → q as false when p is false; a conditional with a false premise is true.
• Confusing inclusive or (∨) with exclusive or (⊕).
• Relying on precedence instead of brackets, especially with → and ↔.
• Forgetting that → is not symmetric: p → q differs from q → p.

Frequently asked questions

How many rows does a truth table with 4 variables have?

2⁴ = 16 rows.

Why is "if false, then anything" true?

The conditional only promises something when the premise is true, so it is not broken when the premise is false.

How do I show two statements are equivalent?

Their result columns must be identical in every row; equivalently, A ↔ B is a tautology.

Sources: OpenStax, Contemporary Mathematics (2023), §2.2 Compound Statements and §2.3 Constructing Truth Tables