A Boolean function of 
variables is a function from
to 
, where 
is a nonnegative integer.
The equivalent encoding 
replaces each input and output value 0 by
(O'Donnell 2021, §1.1).
A Boolean function can be identified with the subset of input assignments on which its value is 1. These subsets
form a Boolean algebra under union,
intersection, and complementation
(Comtet 1974, p. 185). Each Boolean function has a unique
representation, up to order, as a disjunction of
complete products, one for each input assignment
on which the function is 1. The constant false function
corresponds to the empty disjunction. There are 
distinct Boolean functions of
labeled variables,
because each of the 
input assignments can independently be assigned either output value (Comtet 1974,
p. 187).
Shannon (1938) proved that a two-valued Boolean algebra (whose members are most commonly denoted 0 and 1, or false
and true) can describe the operation of two-valued electrical
switching circuits. The following table gives the truth
table for the 
possible Boolean functions of two Boolean variables.
 |  |  |  |  |  |  |  |  |  |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
 |  |  |  |  |  |  |  |  |  |
| 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
The names and symbols for these functions are given in the following table (Simpson 1987, p. 539).
XNOR 
A Boolean function is called monotone increasing if 
whenever 
for every input coordinate 
. Identifying an input with the subset
of coordinates whose values are 1, its minimal true inputs form an antichain
of subsets of 
. Conversely, every such antichain
determines a unique monotone increasing Boolean function, including the empty antichain for the constant false function. Counting these
functions on labeled variables is Dedekind's
problem. Boolean functions can also be viewed as assignments of two labels to
the vertices of an 
-hypercube graph.
If Boolean functions that differ only by a permutation of variables are identified, the numbers of inequivalent
monotone increasing Boolean functions for 
, 1, 2, ... are 2, 3, 5, 10, 30, ... (Biswas and Sarkar 2025;
OEIS A003182).
Let 
denote the number of distinct monotone increasing Boolean functions of 
labeled variables having 
minimal true inputs, where 
and 
are nonnegative integers.
Equivalently, 
counts antichains of 
subsets of an 
-element set. Then
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
