Zhegalkin algebra

In mathematics, Zhegalkin algebra is a set of Boolean functions defined by the nullary operation taking the value $$1$$, use of the binary operation of conjunction $$\land$$, and use of the binary sum operation for modulo 2 $$\oplus$$. The constant $$0$$ is introduced as $$1 \oplus 1 = 0$$. The negation operation is introduced by the relation $$\neg x = x \oplus 1$$. The disjunction operation follows from the identity $$x \lor y = x \land y \oplus x \oplus y$$.

Using Zhegalkin Algebra, any perfect disjunctive normal form can be uniquely converted into a Zhegalkin polynomial (via the Zhegalkin Theorem).

Basic identities

 * $$x \land ( y \land z) = (x \land y) \land z$$, $$x \land y = y \land x$$
 * $$x \oplus ( y \oplus z) = (x \oplus y) \oplus z$$, $$x \oplus y = y \oplus x$$
 * $$x \oplus x = 0$$
 * $$x \oplus 0 = x$$
 * $$x \land ( y \oplus z) = x \land y \oplus x \land z$$

Thus, the basis of Boolean functions $$\bigl\langle \wedge, \oplus, 1 \bigr\rangle$$ is functionally complete.

Its inverse logical basis $$\bigl\langle \lor, \odot, 0 \bigr\rangle$$ is also functionally complete, where $$\odot$$ is the inverse of the XOR operation (via equivalence). For the inverse basis, the identities are inverse as well: $$0 \odot 0 = 1$$ is the output of a constant, $$\neg x = x \odot 0$$ is the output of the negation operation, and $$x \land y = x \lor y \odot x \odot y$$ is the conjunction operation.

The functional completeness of the these two bases follows from completeness of the basis $$\{\neg, \land, \lor\}$$.