# Truth table generator

> Truth table of a Boolean expression in up to 6 variables, with the canonical sum of products, product of sums, minterms and maxterms.

Interactive version: https://www.calcopenly.com/math/truth-table-generator
Subject: Math calculators

A truth table lists the value of a Boolean expression for every combination of its inputs: 2ⁿ rows for n variables, so 8 rows for A, B and C. Each row where the expression is 1 is a minterm, and the OR of those minterms is the canonical sum of products; each row where it is 0 is a maxterm, and the AND of the maxterms is the canonical product of sums. Both describe the same function.

Digital logic and computer science students use it to check circuit designs, simplify conditions and verify logic laws. The default, (A and B) or not C, is true in 5 of 8 rows, minterms 0, 2, 4, 6 and 7, and false in rows 1, 3 and 5.

Up to 6 variables (64 rows) are accepted. Precedence runs not, and/nand, xor, or/nor, implies (grouped from the right), then iff; symbols such as !, &, | and ^ also work. An expression true in every row is a tautology, and one false in every row is a contradiction.

## Inputs

- **Boolean expression**: Operators: not (! ~ ¬ or a trailing '), and (& ·), nand, xor (^ ⊕), or (| +), nor, implies (->), iff (<->). Constants 0 and 1.

## Results

- Canonical sum of products — main result
- Canonical product of sums
- Minterms (rows that are 1)
- Maxterms (rows that are 0)
- Rows that are true
- Classification

## Formula

$$
\begin{aligned} F &= \sum_{F(m)=1} m \quad \text{(sum of minterms)} \\[4pt] &= \prod_{F(M)=0} M \quad \text{(product of maxterms)} \end{aligned}
$$

## Worked examples

### A and B

- Boolean expression: A and B
- **Canonical sum of products: A·B**
- **Minterms (rows that are 1): Σm(3)**
- **Rows that are true: 1**
- **Classification: Contingent**
- Checked against: Python itertools.product over (A, B): only A=1, B=1 is true

### A xor B

- Boolean expression: A xor B
- **Canonical sum of products: A'·B + A·B'**
- **Minterms (rows that are 1): Σm(1, 2)**
- **Canonical product of sums: (A + B)·(A' + B')**
- Checked against: Python itertools.product with a != b

### A implies B

- Boolean expression: A -> B
- **Minterms (rows that are 1): Σm(0, 1, 3)**
- **Maxterms (rows that are 0): ΠM(2)**
- **Canonical product of sums: (A' + B)**
- Checked against: Python itertools.product with (not a) or b

### Tautology: A or not A (edge case)

- Boolean expression: A or not A
- **Classification: Tautology**
- **Canonical product of sums: 1**
- **Rows that are true: 2**
- Checked against: Law of excluded middle; Python check over A in (0, 1)

### (A and B) or not C

- Boolean expression: (A and B) or not C
- **Minterms (rows that are 1): Σm(0, 2, 4, 6, 7)**
- **Maxterms (rows that are 0): ΠM(1, 3, 5)**
- **Rows that are true: 5**
- Checked against: Python itertools.product over (A, B, C) with (a and b) or not c

### nand binds tighter than nor

- Boolean expression: A nand B nor C
- **Minterms (rows that are 1): Σm(6)**
- **Canonical sum of products: A·B·C'**
- Checked against: Python: not ((not (a and b)) or c) over itertools.product

## Questions

### How do you make a truth table?

List every combination of the inputs by counting in binary: 2 variables give 4 rows, 3 give 8, and 6 give 64. Then evaluate the expression in each row, working from the innermost operation outward. For (A and B) or not C, the row A = 1, B = 1, C = 1 gives (1 and 1) or 0 = 1, while A = 0, B = 0, C = 1 gives 0 or 0 = 0.

### What are minterms and maxterms?

A minterm is an AND of every variable, each plain or negated, that is 1 in exactly one row; a maxterm is an OR of every variable that is 0 in exactly one row. Row numbers index them: for A, B and C, minterm 6 is A·B·C′ and maxterm 1 is (A + B + C′). A function is the OR of its minterms and, equally, the AND of its maxterms.

### What is the difference between sum of products and product of sums?

A sum of products (SOP) ORs together AND terms, one for each row where the function is 1; a product of sums (POS) ANDs together OR terms, one for each row where it is 0. A xor B is A′·B + A·B′ as a sum of products and (A + B)·(A′ + B′) as a product of sums. Both are complete; the shorter one is whichever has fewer rows to cover.

### What is a tautology in logic?

An expression that is true in every row of its truth table, whatever the inputs. A or not A, the law of excluded middle, is the simplest; (A → B) or (B → A) is another. The opposite, false in every row, is a contradiction, such as A and not A, and anything true in some rows but not others is called contingent.

### When is A implies B true?

A → B is false only when A is true and B is false; in the other three rows it is true. Its truth table therefore has minterms 0, 1 and 3 and a single maxterm, 2, and it is equivalent to not A or B and to its contrapositive, not B → not A. A chain such as A → B → C groups from the right, as A → (B → C).

### How accurate is the truth table generator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 8 worked examples whose answers come from independent sources; for example, “A and B” is checked against Python itertools.product over (A, B): only A=1, B=1 is true.

### Where does the method come from?

Wolfram MathWorld — Truth Table; Wikipedia — Canonical normal form (minterms and maxterms).

## Sources

- [Wolfram MathWorld — Truth Table](https://mathworld.wolfram.com/TruthTable.html)
- [Wikipedia — Canonical normal form (minterms and maxterms)](https://en.wikipedia.org/wiki/Canonical_normal_form)