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BooleanCountingFunction

 BooleanCountingFunction[kmax, n]represents a Boolean function of n variables that gives True if at most kmax variables are True. BooleanCountingFunction[{k}, n]represents a function of n variables that gives True if exactly k variables are True. BooleanCountingFunction[{kmin, kmax}, n]represents a function that gives True if between kmin and kmax variables are True. BooleanCountingFunction[{{k1, k2, ...}}, n]represents a function that gives True if exactly ki variables are True. BooleanCountingFunction[spec, {a1, a2, ...}]gives the Boolean expression in variables ai corresponding to the Boolean counting function specified by spec. BooleanCountingFunction[spec, {a1, a2, ...}, form]gives the Boolean expression in the form specified by form.
At most two conditions are true:
Convert to a disjunctive normal form:
At most two conditions are true:
 Out[1]=
Convert to a disjunctive normal form:
 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 1, 4 or 5 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 which is true when the number of true arguments is k modulo m:
When k=0 and m=2 then we get Xnor:
When k=1 and m=2 then we get Xor:
For other values of k and m we 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:
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:
Nor:
Xor:
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:
BooleanCountingFunction for when exactly i variables are true have disjoint truth sets:
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