Editing Truth function (section)

=== Arity ===
{{See also|arity}}

A concrete function may be also referred to as an ''operator''. In two-valued logic there are 2 nullary operators (constants), 4 [[unary operation|unary operators]], 16 [[binary operation|binary operators]], 256 [[ternary operation|ternary operators]], and <math>2^{2^n}</math> ''n''-ary operators. In three-valued logic there are 3 nullary operators (constants), 27 [[unary operation|unary operators]], 19683 [[binary operation|binary operators]], 7625597484987 [[ternary operation|ternary operators]], and <math>3^{3^n}</math> ''n''-ary operators. In ''k''-valued logic, there are ''k'' nullary operators, <math>k^k</math> unary operators, <math>k^{k^2}</math> binary operators, <math>k^{k^3}</math> ternary operators, and <math>k^{k^n}</math> ''n''-ary operators. An ''n''-ary operator in ''k''-valued logic is a function from <math>\mathbb{Z}_k^n \to \mathbb{Z}_k</math>. Therefore, the number of such operators is <math>|\mathbb{Z}_k|^{|\mathbb{Z}_k^n|} = k^{k^n}</math>, which is how the above numbers were derived.

However, some of the operators of a particular arity are actually degenerate forms that perform a lower-arity operation on some of the inputs and ignore the rest of the inputs. Out of the 256 ternary Boolean operators cited above, <math>\binom{3}{2}\cdot 16 - \binom{3}{1}\cdot 4 + \binom{3}{0}\cdot 2</math> of them are such degenerate forms of binary or lower-arity operators, using the [[inclusion–exclusion principle]]. The ternary operator <math>f(x,y,z)=\lnot x</math> is one such operator which is actually a unary operator applied to one input, and ignoring the other two inputs.

[[Negation|"Not"]] is a [[unary operation|unary operator]], it takes a single term (¬''P''). The rest are [[binary operation|binary operators]], taking two terms to make a compound statement ({{math|''P'' &and; ''Q'',{{wrap}}''P'' &or; ''Q'',{{wrap}}''P'' → ''Q'',{{wrap}}''P'' ↔ ''Q''}}).

The set of logical operators {{math|&Omega;}} may be [[Partition of a set|partitioned]] into disjoint subsets as follows:

::: <math>\Omega = \Omega_0 \cup \Omega_1 \cup \ldots \cup \Omega_j \cup \ldots \cup \Omega_m \,.</math>

In this partition, <math>\Omega_j</math> is the set of operator symbols of ''[[arity]]'' {{mvar|j}}.

In the more familiar propositional calculi, <math>\Omega</math> is typically partitioned as follows:

:::nullary operators: <math>\Omega_0 = \{\bot, \top \} </math>

:::unary operators: <math>\Omega_1 = \{ \lnot \} </math>

:::binary operators: <math>\Omega_2 \supset \{ \land, \lor, \rightarrow, \leftrightarrow \} </math>