At most two conditions are true:

Convert to a disjunctive normal form:

Specify thatf is true when at most2 arguments are true:

Exactly2 arguments are true:

Between2 and3 arguments are true:

1,3, or5 arguments are true:

Specify thatf is true when exactly1,4, or5 arguments are true:

BooleanCountingFunction is by default preserved in function form:

UseBooleanConvert 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:

Create new primitives that are true when at most, at least, or exactlyk 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 isk modulom:

Whenk=0 andm=2, you getXnor:

Whenk=1 andm=2, you getXor:

For other values ofk andm, 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:

Find the mean time to failure for a system that needs two out of three components to work:

BooleanCountingFunction is symmetric in its arguments:

Logical combinations ofBooleanCountingFunction correspond to set operations on indices:

The basic specification can equivalently be specified usingRange:

Many primitives can be expressed in terms ofBooleanCountingFunction:

And:

Or:

Nand:

Nor:

Xor:

Xnor:

Equivalent:

Majority:

The size of the truth set forBooleanCountingFunction is the length ofSubsets:

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

BooleanCountingFunction for when exactlyi variables are true has disjoint truth sets: