Boolian Algebra [Three Lessons]
| Spring 1887 | Houghton Library |
In this system of signs, each letter of the alphabet is the abbreviated statement of a fact, simple or complex. Thus, x might be taken to signify that twice two is four, and that either all men are mortal or else Napoleon Bonaparte was born in Salem, Mass. Every letter is an abbreviated statement. It may be of all that is stated in a book, it may be of something very simple.
The sign = is used in ordinary algebra to signify equality. Thus, x = y, read “x equals y,” can be written when the quantity named x is known to be equal to the quantity named y. In the Boolian Algebra the same sign is used, and is read in the same words, but a different meaning is attached to the sign and to the words “equal,” etc. with which it is associated. Two facts are considered to be equal when every possible state of things in which either is true is or would be a state of things in which the other is or would be true, and vice versa. Thus, “this polygon has three sides,” and “this polygon has three angles” are equal or equivalent facts, although the statements refer the one to sides, the other to angles. If you ask what is meant by a “possible” state of things, I reply that the possible is that which is not known to be non-existent,—and that either in the state of knowledge in which we find ourselves, or else in some feigned condition of ignorance. The “possible” of this Algebra may be any variety of the possible,—as the logically possible, the physically possible, etc. Only the same meaning must be attached to the possible throughout any one discussion. Or if different varieties are used, these must be carefully distinguished. It must be understood that when x = y is written the facts x and y are not stated; it is only stated that they are either both actual or neither.
The sign + is used in ordinary algebra to denote addition. Thus, x + y, read “x plus y,” denotes the sum of the quantities named x and y. This sign, with the words “plus,” “sum,” “add,” etc. associated with it, are also used in the Boolian Algebra, but by most writers (myself included) in a sense not strictly analogous to the arithmetical sense. The expression x + y is a statement, and it states that either x or y (perhaps both) is a fact, without saying which. The expression x + y is that statement which is true if either of the statements x and y is true, and is false if both x and y are false. It follows that x + x = x, contrary to the analogy of arithmetic.
The signs × and ·, and the writing one after another of two expressions, are used in ordinary algebra to denote multiplication. Thus, x × y, or x · y, or xy, read “xy, ” or “x into y,” or “x multiplied by y” denote the product of x and y. The same signs and words are used in the Boolian Algebra in a different sense. The expression xy is a statement, and it is the statement of both facts x and y. More precisely, it is that statement which is true if both the statements x and y are true, and is false if either of them is false. It follows that xx = x, which in ordinary algebra would be an equation satisfied by x = 0, x = 1, and x = ∞.
Exercise. Ascertain whether the following formulae necessarily hold good, whatever statements x, y, and z may be.
x + y = y + x xy = yx x(yz) = (xy)z
[N.B. Parentheses are used to signify that the statement in parenthesis is to be combined as one statement by addition or multiplication with the one outside of the parenthesis.]
x + (y + z) = (x + y) + z
x(y + z) = xy + xz
x + yz = (x + y)(x + z)
(x + y)(y + z)(z + x) = xy + yz + zx
N.B. The exercises should be done, and the formulae verified by the definitions of logical addition and multiplication in the way shown in the following example: x + yz is true if either x or yz is true; otherwise it is false. But yz is only true if both y and z are true. Hence x + yz is true 1st if x is true and 2nd if both y and z are true. The statement (x + y)(x + z) is true only if (x + y) and (x + z) are true, x + y is true 1st if x is true, and 2nd if y is true, x + z is true, 1st if x is true, and 2nd if z is true. Hence both (x + y) and (x + z) are true, 1st if x is true, and 2nd if both y and z are true. But these are the same circumstances under which x + yz is true. Hence, the two statements x + yz and (x + y)(x + z) are equivalent, and we can write x + yz = (x + y)(x + z). Do all these exercises before going further.
Boolian Algebra—Second Lesson
The equations x + (y + z) = (x + y) + z and x(yz) = (xy)z are called respectively the Associative Principle of Addition and the Associative Principle of Multiplication. The formulae x + y = y + x and xy = yx are called the Commutative Principle of Addition and Multiplication. The formula x(y + z) = xy + xz is called the Distributive Principle of Multiplication with respect to addition. The formula x + yz = (x + y)(x + z) is called the Distributive Principle of Addition with respect to multiplication.
From those formulae with x + x = x and xx = x all others can be deduced, without recourse to the meanings of the operations. Thus, to prove (a + b)(c + d) = ac + ad + bc + bd we proceed as follows. By the distributive principle (a + b)(c + d) = (a + b)c + (a + b)d. By the commutative principle this equals c(a + b) + d(a + b) and, again applying the distributive principle, this equals ca + cb + da + db or by the commutative principle ac + bc + ad + bd. The associative principle is assumed in leaving off the parentheses.
Exercises. Prove in this way the following.
ab + cd = (a + c)(a + d)(b + c)(b + d)
a + ab = a
a(a + b) = a
(a + b)(b + c)(c + a) = ab + bc + ca
(a + b)(b + c)(c + d)(d + a) = ac + bd
(a + b + c)(a + b + d)(a + c + d)(b + c + d) = ab + ac + ad + bc + bd + cd
(a + b + c)(a + b + d)(b + c + e)(c + a + f)(a + d + f)(b + d + e) (c + e + f)(d + e + f) = ae + bf + cd
(a + e)(b + f)(c + d) = abc + abd