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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.

Drafted with AI writing toolsUpdated Checked examples: 8

Operators: not (! ~ ¬ or a trailing '), and (& ·), nand, xor (^ ⊕), or (| +), nor, implies (->), iff (<->). Constants 0 and 1.
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Canonical sum of products
A'·B'·C' + A'·B·C' + A·B'·C' + A·B·C' + A·B·C
Canonical sum of products: A'·B'·C' + A'·B·C' + A·B'·C' + A·B·C' + A·B·C
Canonical product of sums
(A + B + C')·(A + B' + C')·(A' + B + 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
Classification
Contingent

The expression is true in 5 of 8 rows (minterms 0, 2, 4, 6, 7), so its value depends on the inputs.

Columns of the truth table (row 0 on the left)

A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
F
1
0
1
0
1
0
1
1
Truth table Rows: 8
RowABCF
00001
10010
20101
30110
41001
51010
61101
71111
How it's calculated S

  1. Read with operator precedence

    (A∧B)∨¬C\left(A \land B\right) \lor \lnot C

    not binds tightest, then and/nand, xor, or/nor, implies (grouped from the right), and iff last.

  2. Evaluate all 8 rows

    Each row sets A, B, C to one combination of 0s and 1s, counting in binary with A as the most significant bit; the row number is the minterm index.

  3. Minterms and maxterms

    F=∑m(0,2,4,6,7)=∏M(1,3,5)F = \sum m(0, 2, 4, 6, 7) = \prod M(1, 3, 5)

  4. Canonical sum of products

    F=A‾B‾C‾+A‾BC‾+AB‾C‾+ABC‾+ABCF = \overline{A} \overline{B} \overline{C} + \overline{A} B \overline{C} + A \overline{B} \overline{C} + A B \overline{C} + A B C

    One product term for each row where F = 1; a bar means the variable is 0 in that row.

  5. Canonical product of sums

    F=(A+B+C‾)(A+B‾+C‾)(A‾+B+C‾)F = (A + B + \overline{C})(A + \overline{B} + \overline{C})(\overline{A} + B + \overline{C})

    One sum term for each row where F = 0; each term is 0 only in its own row.

About the truth table generator

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.

How to make a truth table

List every combination of the inputs by counting in binary, then evaluate the expression in each row. Two variables give 2² = 4 rows (00, 01, 10, 11), three give 8, and n variables give 2ⁿ. For A and B, only row 11 is true; for A or B, every row except 00 is true.

Evaluate from the innermost parentheses outward. For (A and B) or not C in the row A = 1, B = 1, C = 0, the bracket gives 1 and not C gives 1, so the row is 1.

Truth tables of the basic operators

Each pattern lists the output for the rows AB = 00, 01, 10 and 11, in that order:

  • A and B: 0001, true only when both inputs are true.
  • A or B: 0111, true when either or both are true. This is inclusive or.
  • A xor B: 0110, true when exactly one is true.
  • A nand B: 1110, the opposite of and.
  • A nor B: 1000, the opposite of or.
  • A implies B: 1101, false only when A is true and B is false.
  • A iff B: 1001, true when the inputs match.

The negation not A has the pattern 1100 over the same rows, because it simply flips A.

Worked expressions and their minterms

ExpressionMintermsClassification
A and BΣm(3)Contingent
A xor BΣm(1, 2)Contingent
A implies BΣm(0, 1, 3)Contingent
A or not AΣm(0, 1)Tautology
A and not AΣm()Contradiction
(A and B) or not CΣm(0, 2, 4, 6, 7)Contingent

A minterm number is the row read as a binary number, with the first variable as the most significant bit. (A and B) or not C is true in rows 0 (000), 2 (010), 4 (100) and 6 (110), where C is 0, and in row 7 (111), where A and B are both 1.

Operator precedence

Without parentheses, not binds first, then and/nand, then xor, then or/nor, then implies, and iff last. So A or B and C means A or (B and C), true in rows 3 to 7: Σm(3, 4, 5, 6, 7). (A or B) and C is true only in rows 3, 5 and 7.

Implication groups from the right: A → B → C means A → (B → C), which is false only in row 6 (110). (A → B) → C is a different function, false in rows 0, 2 and 6.

Textbooks and programming languages do not all rank these operators the same way. Add parentheses whenever you copy a formula from another source.

When is A implies B true?

A → B is false only when A is true and B is false. With a false premise it is true whatever B is, which can surprise readers used to the everyday "if … then". Discrete Mathematics: An Open Introduction explains why logic uses this convention.

Implication has the same truth table as not A or B, and as its contrapositive, not B → not A.

Tautology, contradiction and equivalence

A tautology is true in every row, as A or not A is. A contradiction is false in every row, as A and not A is. An expression true in some rows and false in others is contingent.

To test whether two expressions are equivalent, join them with iff: the result is a tautology exactly when they agree in every row. De Morgan's law passes this test, because not (A and B) iff (not A or not B) is a tautology.

Minterms, maxterms, SOP and POS

A minterm is a row where the output is 1. The canonical sum of products (SOP) ORs one AND-term per minterm, with each variable plain where its value is 1 and primed (negated) where it is 0. For A xor B, the SOP is A'·B + A·B'.

A maxterm is a row where the output is 0. The canonical product of sums (POS) ANDs one OR-term per maxterm, with each variable plain where its value is 0 and primed where it is 1. For A xor B, the POS is (A + B)·(A' + B').

Canonical forms list every row, so they are usually longer than a simplified expression: A or not A is a tautology, yet its SOP is A' + A. See canonical normal form for the naming of minterms and maxterms.

Common mistakes

  • Reading or as exclusive. A or B is also true when both A and B are true; use xor for exactly one.
  • Expecting a false premise to make A → B false. Rows 00 and 01 of implication are both true.
  • Reading ^ as a power. In Boolean expressions ^ means xor and + means or.
  • Numbering rows with a different variable order. Minterm 1 means A = 0, B = 1 only when A is the first column.

Input notes

  • Up to 6 variables, which gives 64 rows. Names use letters, digits and underscores, start with a letter or underscore, and are case-sensitive.
  • The words and, or, not, xor, nand, nor, implies, iff, true and false are reserved. The constants are 0 and 1, or false and true.
  • Symbol forms also work: !, ~, ¬ or a trailing ' for not; & or · for and; | or + for or; ^ or ⊕ for xor; -> for implies; and <-> for iff.
  • Variables are ordered alphabetically, so B and not A still puts A in the first column; its single minterm is row 01, Σm(1).

To apply the same logic to lists of items, use the set operations calculator, where union matches or and intersection matches and. For and, or and xor on the bits of whole numbers, use the bitwise calculator.

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: Independent calculation

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: Independent calculation

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: Independent calculation

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

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).

About this calculator

F=∑F(m)=1m=∏F(M)=0M\begin{aligned} F &= \sum_{F(m)=1} m \\[4pt] &= \prod_{F(M)=0} M \end{aligned}

Sources

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

Checked against references

This calculator includes 8 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.

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