BUILT-IN MATHEMATICA SYMBOL

BooleanCountingFunction

represents a Boolean function of n variables that gives True if at most

variables are True.

represents a function of n variables that gives True if exactly k variables are True.

BooleanCountingFunction

represents a function that gives True if between

and

variables are True.

BooleanCountingFunction

represents a function that gives True if exactly

variables are True.

BooleanCountingFunction

gives the Boolean expression in variables

corresponding to the Boolean counting function specified by spec.

BooleanCountingFunction

gives the Boolean expression in the form specified by form.

MORE INFORMATION

BooleanCountingFunction[spec] gives a Boolean function object that works like Function.

BooleanCountingFunction[spec][a₁, a₂, ...] gives an implicit representation equivalent to the explicit Boolean expression BooleanCountingFunction.

BooleanConvert converts BooleanCountingFunction[spec][vars] to an explicit Boolean expression.

BooleanCountingFunction represents a function that gives True if , , ..., variables are True.

Any symmetric Boolean function can be represented uniquely using BooleanCountingFunction.

In BooleanCountingFunction, the possible forms are as given for BooleanConvert.

BooleanCountingFunction by default gives an expression in DNF.

EXAMPLES

CLOSE ALL

Basic Examples (1)

At most two conditions are true:

Convert to a disjunctive normal form:

At most two conditions are true:

In[1]:=

Out[1]=

Convert to a disjunctive normal form:

In[2]:=

Out[2]=

Scope (6)

Specify that f is true when at most

arguments are true:

Exactly

arguments are true:

Between

and

arguments are true:

,

, or

arguments are true:

Specify that f is true when exactly

,

, or

arguments are true:

BooleanCountingFunction is by default preserved in function form:

Use BooleanConvert to convert to other forms:

BooleanCountingFunction is automatically converted when given an explicit list of variables:

The expanded forms can be large when the number of variables grows:

The performance gain in evaluating the function form can be substantial:

Constant arguments are reduced:

Extreme cases are automatically converted to formulas:

Applications (3)

Create new primitives that are true when at most, at least, or exactly k arguments are true:

Create a number of disk regions along the unit circle:

Show the newly combined regions:

Integrate over these regions:

Define a Boolean function that is true when the number of true arguments is k modulo m:

When k=0 and m=2 you get Xnor:

When k=1 and m=2 you get Xor:

For other values of k and m you get new functionality:

The 2D truth table:

Define a Boolean function that sorts a list of truth values:

The resulting list is always in sorted order:

Properties & Relations (6)

BooleanCountingFunction is symmetric in its arguments:

Logical combinations of BooleanCountingFunction correspond to set operations on indices:

The basic specification can equivalently be specified using Range:

Many primitives can be expressed in terms of BooleanCountingFunction:

And:

Or:

Nand:

Nor:

Xor:

Xnor:

Equivalent:

Majority:

The size of the truth set for BooleanCountingFunction is the length of Subsets:

The size of the truth set for BooleanCountingFunction can be given by a combinatorial sum:

Neat Examples (1)

BooleanCountingFunction for when exactly i variables are true has disjoint truth sets: