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12

[Reply to Loring]

30 December 1887

Houghton Library

Milford, Pa., 1887 Dec. 30.

Mr. J. B. Loring, Box 555 New York.

My dear pupil:

I congratulate you on the way you are taking hold of the subject. I have received yours of Dec. 22 and 27. You will please make it a rule to report to me at the end of each four hours’ work, so that we shall know when the quarter ends; for its length is determined by the amount of work that you have done, measured in time.

I will first consider the equation

(x + y)(y + z)(z + x) = xy + yz + zx.

Your reasoning in your letter is pretty well. It does not fully meet the case. You show that the two statements are both true provided that both x and % are true. But that is not enough. You must also show that neither can be true without the other being true. Besides, as a matter of practice in this system of signs, I want you to prove it symbolically. First, multiply together the first two factors of the first member. That will give by the distributive principle of addition with respect to multiplication (x + y)(y + z) = y + xz. Now multiply in the third factor. That will give

In your second letter, you ask how the distributive principle proves that (a + b)(c + d) = (a + b)c + (a + b)d. The formula of the distributive principle of multiplication with respect to addition is x(y + z) = xy + xz. Put x = a + b, y = c, and z = d, and you have the result. Of course, in the general formula, x, y, z, may be any statements; hence, it is legitimate to adopt the equivalents just given.

The next question is what I mean by saying that the associative principle is assumed in leaving off the parentheses. By the associative principle χ + (y + z) = (x + y) + z; so that we may as well write simply x + y + z, for whether this means that x and y are first to be added and then z added on to them, or that to x is to be added the sum of y and z makes no difference according to the preceding formula. In like manner, in the particular case in which I make the remark and to which your inquiry relates, without the associative principle, I should only reach the statement (ac + bc) + (ad + bd). But by the associative principle, this would be the same as ac + [bc + (ad + bd)] and as ac + [(bc + ad) + bd], and in short, without giving all the equivalents, it obviously makes no difference how the parentheses are put in, so long as the factors of no one term are separated, so that they may as well be dropped altogether. All students have to ask such questions at first.

The blurred lines are x + 0 = x x$ = x 0x = 0 $ + x = $ [This is called the principle of contradiction.] [This is the so-called principle of excluded middle.] In the last two formulae please observe the second x is in each case negatived.

I will not send you any further exercise today, as I think you have enough for 8 hours, at least. These things are puzzling, at first.

Yours very truly,