Fan-out in digital logic: meaning and examples

Q: Does fan-out change the truth table?

No. (A AND B) OR (A AND C) and A AND (B OR C) have the same table, which the distribution law proves on every row. Fan-out changes the shape and the electrical load of the circuit, not the function it computes.

Fan-out is the number of gate inputs that one signal drives. In a logic diagram it is the branching of a wire; in a formula it is a variable or subexpression that appears more than once.

Every time an input or an intermediate result is used by more than one gate, its wire splits, and the count of destinations is its fan-out. Electrically, fan-out is limited, because each input draws current from the output driving it, and a signal that feeds too many gates must be buffered. Logically, fan-out is what distinguishes a tree of gates from a general network: in a tree every signal is used once, and a formula with a repeated variable cannot be drawn as a tree.

The artefact on this page is (A AND B) OR (A AND C), where A feeds both AND gates, so A has a fan-out of two while B and C each drive one gate. The 8-row table reads true on rows 5, 6 and 7. The satisfiable check holds with the witness A = TRUE, B = FALSE, C = TRUE, and the tautology check is violated with the counterexample A = FALSE, B = FALSE, C = FALSE. The distribution law would rewrite the formula as A AND (B OR C), where A is used once.

Fan-out matters in hardware for timing and loading, and in software whenever a condition is repeated: the same sub-check written in two places must be kept in step. The engine's logic diagram draws shared inputs with branching wires, and the simplifier's distribution and absorption laws are often what reduce fan-out. The term is confused with fan-in, the number of inputs a single gate has, and with the branching of a decision tree, which is a different structure.

Example

f := (A AND B) OR (A AND 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 = T, B = F, C = T · Exhaustively checked and solver cross-checked · 6 rows

Can be true? Yes Example assignment: A = T, B = F, C = T · Exhaustively checked and solver cross-checked · 6 rows

Truth table (8 rows)

Fan-out in digital logic: meaning and examples
#	A	B	C	f
0	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	TRUE	FALSE
2	FALSE	TRUE	FALSE	FALSE
3	FALSE	TRUE	TRUE	FALSE
4	TRUE	FALSE	FALSE	FALSE
5	TRUE	FALSE	TRUE	TRUE
6	TRUE	TRUE	FALSE	TRUE
7	TRUE	TRUE	TRUE	TRUE

Plain-language reading

Beginnerf is true exactly when at least one of the following holds (both may hold):
1. both A and B are true
2. both A and C are true
In every other case, f is false.
About the operators:
- OR is inclusive: it is true when at least one part is true, including when several are.
- AND is true only when all of its parts are true.

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

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

Logic diagram

The common slip

Treating fan-out as a property of the function. The diagram follows the formula as written: (A AND B) OR (A AND C) shows A branching to two gates, and the equivalent A AND (B OR C) shows it feeding one. Same table, different fan-out; the number belongs to the drawing.

Questions

Does fan-out change the truth table?

No. (A AND B) OR (A AND C) and A AND (B OR C) have the same table, which the distribution law proves on every row. Fan-out changes the shape and the electrical load of the circuit, not the function it computes.

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.