Cofactor (Boolean Algebra)

boolean-algebra

Definition

Cofactor

In Boolean Algebra, the term “cofactor” refers to a derived Boolean function obtained by assigning a fixed value to a specific variable within a Boolean function.

For a given Boolean function $f(x_1,x_2,\dots ,x_n)$ and a variable $x_i$, the cofactors are defined as follows:

Positive Cofactor:

$$f_{x_i=1}(x_1,\dots ,x_{i-1},x_{i+1},\dots ,x_n)=f(x_1,\dots ,x_{i-1},1,x_{i+1},\dots ,x_n)$$

Negative Cofactor:

$$f_{x_i=0}(x_1,\dots ,x_{i-1},x_{i+1},\dots ,x_n)=f(x_1,\dots ,x_{i-1},0,x_{i+1},\dots ,x_n)$$

Cofactors can be used for analysing and simplifying Boolean functions using the Shannon expansion.