Truth table: meaning and examples

A truth table lists every combination of values the inputs of a formula can take, one row per combination, and the value the formula gives on each. With n inputs it has 2 to the power n rows.

A truth table is the complete, finite description of a Boolean function: nothing about the function is left out, so any property can be read off it by inspection. The rows follow a fixed order. In this reference, row 0 is the all-FALSE row and the first input is the most significant bit, so the rows count upward in binary. The columns show the inputs, any assumptions, and one column per output; in argument mode they show each premise and the conclusion instead. A truth table is also called a table of values or, in older texts, a matrix.

The table on this page is for (A OR B) AND NOT C: three inputs, 8 rows. The output column reads false, false, true, false, true, false, true, false, so f is true on rows 2, 4 and 6, the rows where C is false and at least one of A and B is true. The checks read the same rows: the tautology check is violated with the counterexample A = FALSE, B = FALSE, C = FALSE, and the satisfiable check holds with the witness A = FALSE, B = TRUE, C = FALSE.

Truth tables are the ground truth behind everything else in this reference: equivalence, validity, normal forms and Karnaugh maps all reduce to comparing or reading columns. In hardware they are the specification of a combinational circuit; in testing they enumerate the cases a rule must handle; in a philosophy course they decide validity. The term is confused with a decision table, which lists only the cases that matter and may leave combinations out.

Example

f := (A OR B) AND NOT C · output

Exact checks

Always true (tautology)? No Counterexample: A = F, B = F, C = F · Exhaustively checked and solver cross-checked · 1 rows

Always false (contradiction)? No Counterexample: A = F, B = T, C = F · Exhaustively checked and solver cross-checked · 3 rows

Can be true? Yes Example assignment: A = F, B = T, C = F · Exhaustively checked and solver cross-checked · 3 rows

Truth table (8 rows)

Truth table: meaning and examples
#	A	B	C	f
0	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	TRUE	FALSE
2	FALSE	TRUE	FALSE	TRUE
3	FALSE	TRUE	TRUE	FALSE
4	TRUE	FALSE	FALSE	TRUE
5	TRUE	FALSE	TRUE	FALSE
6	TRUE	TRUE	FALSE	TRUE
7	TRUE	TRUE	TRUE	FALSE

Plain-language reading

Beginnerf is true exactly when both of the following hold:
1. at least one of A or B is true (both may be true)
2. C is false
In every other case, f is false.
About the operators:
- AND is true only when all of its parts are true.
- OR is inclusive: it is true when at least one part is true, including when several are.
- NOT reverses a value: true becomes false and false becomes true.

Intermediatef is true exactly when both of the following hold:
1. at least one of A or B is true (both may be true)
2. C is false
In every other case, f is false.

Advancedf is true exactly when both of the following hold [conjunction ∧]:
1. at least one of A or B is true (both may be true) [inclusive disjunction ∨]
2. C is false [negation ¬]
In every other case, f is false.
Formal notation: (A ∨ B) ∧ ¬C

The common slip

Assuming everyone numbers rows the same way. Some textbooks start with the all-TRUE row, or make the last variable the most significant bit. A Σm list copied between conventions names different rows, so check the convention before comparing tables.

Questions

How big can a truth table get?

Each extra input doubles the rows: 2 inputs give 4 rows, 10 give 1,024, 20 give over a million. The engine shows tables for up to 12 inputs and runs its exhaustive checks up to 20; beyond that it uses a SAT solver instead of a table.

Is a truth table a proof?

Yes, for propositional logic. A property checked on every row is proved by truth table, because the rows are all the cases there are. That is what the exact checks on this page do.

Guides

How to make a truth table

Practice

Glossary

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.