Boole's expansion theorem

Boole's expansion theorem, often referred to as the Shannon expansion or Shannon decomposition, is the identity F = x ⋅ F x + x ′ ⋅ F x ′ {\displaystyle F=x\cdot F_{x}+x'\cdot F_{x'}} , where F {\displaystyle F} is any Boolean function, x {\displaystyle x} is a variable, x ′ {\displaystyle x'} is the complement of x {\displaystyle x} , and F x {\displaystyle F_{x}} and F x ′ {\displaystyle F_{x'}} are F {\displaystyle F} with the argument x {\displaystyle x} set equal to 1 {\displaystyle 1} and to 0 {\displaystyle 0} respectively. The terms F x {\displaystyle F_{x}} and F x ′ {\displaystyle F_{x'}} are sometimes called the positive and negative Shannon cofactors, respectively, of F {\displaystyle F} with respect to x {\displaystyle x} .

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