Note on the Condition that a Boolean Equation Have a Unique SolutionAuthor(s): B. A. BernsteinSource: American Journal of Mathematics, Vol. 54, No. 2 (Apr., 1932), pp. 417-418Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371005 .

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NOTE ON THE CONDITION THAT A BOOLEAN EQUATION HAVE A UNIQUE SOLUTION.*

By B. A. BERNSTEIN.

Consider the general Boolean equation

(i) f (x, y, *, t)-Axy * * t + Bxy . D' + + Lx"y' * ,

involving n -unknowns x, y, * * , t and having A, B, ,L for its 2n dis- criminants. Consider also the equations

(ii) x=a, y =b, , t 7e and the equation

(iii) + (x, y, .. *, t) =-a.'x -+ ax' + b'y + by' + * + 7gt + let' =-O.

Whitehead . has shown that the necessary and sllfficient condition that (i) have the unique solution (ii) is that (i) be of the form (iii). Professor Whitehead's proof, carried out only for an equation in two unknowns, con- sists in showing that the necessary and sufficient condition that (i) have the unique solution (ii) is relation

(iv) AB . * L + A'B' + A"C'C + - + K'L' Op

which relation he proved earlier to be a necessary and sufficient condition that f of (i) be of the :form 0 of (iii). But Professor Whitehead's pro'ofs require a good bit of calculation, especially for the general case of n variables.? The object of this note is: (1) to offer a very simple proof of the fact that an equation of form (iii) has the solution (ii) an-d conversely; (2) to offer a very simple proof of the fact that (iv) is the condition that f of (i) is of the form 0 of (iii) ; and (3) to call attention to the simple geometry uinder- lying condition (iv).

* Presented to the Society, April 11, 1931. t The usual Boole-Schroder iiotation is used, except that a' dlenotes the negative

of a. : A. N. Whitehead, " Memoir on the Algebra of Symbolic Logic," American Journal

of Mathematics, Vol. 23 (1901), pp. 140-150. ? To obtain (iv) as the necessary condition that (i) have a unique solution

Whitehead eliminates from (i) all the variables except one, in turn, then gets the conditions that the resulting equations have unique solutions, and then combines these conditions into a single condition. To obtain (iv) as the condition that f of (i) be of the form 4 of (iii), Professor Whitehead develops f normally with respect to the variables, then identifies corresponding discriminants, and then simplifies.

417

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418 B. A. BERNSTEIN.

1. A very simple proof of the fact that an equation of form (iii) has the solution (ii) and conversely, consists in merely observing that equation (iii) and the system of equations (ii) are each equivalent to the system

(v) a'x + ax' O, b'y + by' =0O * , k't + kt-'-O.

It will be noted that this proof does not use (iv) at all.

2. To obtain (iv) very simply as the necessary and sufficient condition that f of (i) be of the form ( of (iii), observe that the discriminants of ( are

+(, , * 1) a' + b' + * +k', 0(,1, ,~0) a+ **+k (O, O,* * a+ b + *+

Hence, f will be identical with ( when and only when

A -a'`+b'`+ * *+ k`, B '+ b + * *+ k,P .. *, L a +b + -+k, or (vi) A' ab . , B'== ab . . . , ** , Lo a"b'* * * k'.

But the right-hand members of (vi) are seen to be the constituoents in the normal development of 1 with respect to a, b, . , k . Hence,

A' + B" + + L` =-L 1, ATB' ==O, A'C' =O, . , K'L' O, or (iv) AB * L + A'B + AC'K + * * +K'L'= O.

Of course, since (ii) is equivalent to (iii), we have that (iv) is also the condition that (i) have the solution (ii).

3. The simple geometry underlyinig condition (iv) is seen from (vi). The necessary and sufficient condition that (i) have the solution (ii), or that f of (i) be of the form (p of (iii), is, geometrically, that the regions representing the negatives of the discriminants of f be the regions corre- sponding to the 2n constituents in the normal development of 1 with respect to a, b, * , kl of (p. The geometry for the general equation (i) is thus a very simple extension of the geometry for the equation in one unknown, a'x + 4x' 0.

UNIVE,RSITY OF CALIFORNIA,

BERKELEY, CALIFORNIA.

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