By denoting Øp by , and pÙq by pq, working in line segment notation, with square-free polynomials, and interpreting unmarked state as 0, various tautologies are easy to prove. For example,

The proposed method is very powerful even in direct proofs of more complex tautologies, e.g., a crosstransposition

The left side gives:

The right side results in:

so this proves the tautology.

Duality (De Morgan laws) holds for Ù and Ú, after replacing 0 by 1 and vice versa, and each variable p by its inverse . For example, a dual tautology for the crosstransposition

can be proved in the same way, working with =1-p and pÚq=p+q-pq. All proofs and calculations hold in the extended continuous set [0,1], i.e., in the polyvalent Boolean logic.

In order to simplify, we can calculate square-free polynomials corresponding to other logic operations and work with them in the same way as before. For example, pÞ q = 1 - p + pq, pÛ q=1 - p - q + 2pq, pq = p+q-2pq, etc. The most interesting for a future use in logical gates can be the Sheffer stroke NAND, p|q=[`pq], that can be algebraically expressed as p|q=1-pq. Operations (Ø, Ù), or (Ø, Ú) are a base of Boolean polyvalent logic: it is possible to express all other logical operations by them. Sheffer operation NAND (| or in his original notation) and its dual, Lukasiewicz's operation NOR (¯ ), are single-operation logical bases.

After reducing a complicated logical expression and obtaining corresponding square-free polynomial, we can be interested to find from it a simplified logical expression. For this, it is sufficient to take all possible values for its variables that belong to the discrete set {0,1} and write the corresponding conjuctive or disjunctive normal form (CNF or DNF). For example, for the reduced square-free polynomial 1-p+pq, by taking for (p,q) values from the set {(0,0),(0,1),(1,0),(1,1)} we obtain DNF ØpÚq.