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 = $
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,
[Additional Exercises in Boolian Algebra]
| Summer 1887 | Houghton Library |
I saw a man in the street who had a way of looking up and squinnying his eyes which showed me that he was either excessively nearsighted or somewhat foolish. He went up to a post-box at the corner and tried to put some object through the top, until he finally found the slit in the side of the box. He was not foolish enough to account for this conduct, and was certainly not intoxicated; so that he was plainly either excessively near-sighted or else absent-minded. Having put something small into the box he next sat down on the curbstone. Nearsightedness would not account for this, which showed him to be either absent-minded or rather foolish. But absentmindedness would not account for his squinnying his eyes; so that I was puzzled. He might be both absent-minded and somewhat foolish, and if so undoubtedly was under the impression he was in a bob-tailed horse-car, and had deposited a nickel in the box. If he was somewhat foolish and excessively near-sighted, the same conclusion would result. If he was absent-minded and excessively near-sighted, the inference will be the same.
Let a mean he was absent-minded; f, that he was foolish; n, that he was excessively near-sighted; c, that he thought himself in a horse-car. Then the premises are
| (a + f), | he was absent-minded or rather foolish; |
| (f + n), | he was rather foolish or excessively near-sighted; |
| (n + a), | he was very near-sighted or else absent-minded; |
|
|
if he was both absent-minded and foolish, he thought himself in a bob-tailed horse-car; |
|
|
if he was both foolish and near-sighted, he thought himself in a bob-tailed horse-car; |
|
|
if he was both near-sighted and absent-minded, he thought himself in a bob-tailed horse-car. |
The product of the premises is
He must have thought himself in a bob-tailed horse-car.
1. Wherever there is smoke, there is fire or light. Wherever there is light and smoke there is fire. There is no fire without either smoke or light. What if there is smoke? What if there is not smoke? What if there is no light?
2. The members of a board were all either bondholders or shareholders, but no member was both bondholder and shareholder; and the bondholders were all on the board. State the relation between bond-holders and shareholders.
3. Three persons are set to sort a heap of books. The first is to collect all English political works and bound foreign novels; the second, the bound political works and English novels not political; the third, the bound English works and unbound political novels. What books will be claimed by two and by all of them?
4. The members of a scientific society are divided into three sections, which are denoted by A, B, C. Every member must join one, at least, of these sections, subject to the following conditions: 1st, whoever is a member of A but not of B, of B but not of C, or of C but not of A, may read a paper if he has paid his subscription, but otherwise not; 2nd, whoever is a member of A but not of C, of C but not of A, or of B but not of A, may exhibit an experiment if he has paid his subscription but otherwise not; 3rd, every member must either read a paper or exhibit an experiment annually. Find the least addition to the rules necessary to compelling every member to pay his yearly subscription.
5. In a certain lot of calicos, every piece with lilac spots and green spots has also black spots and yellow spots, and vice versa; and every piece with red spots and orange spots has also blue spots and yellow spots, and vice versa. Eliminate yellow and express the conclusion in terms of green.
6. In the Kingdom of Mbugwam, all freemen together with all cannibal cooks and slaves neither cannibals nor cooks are attached either to the army or the court. The army cooks not attached to the court as well as all men in the army attached to the court except cooks are, so many as are cannibals, slaves. The cooks and slaves belonging to the army are in attendance at court if they are not cannibals but not if they are; while all other men are in attendance at court if they are cannibals, but not if they are not. Without distinguishing slaves from freemen, 1st state the composition of the army, 2nd state of what classes the cooks consist. 3rd, without distinguishing either slaves from freemen or men in the army from men out of it, describe the cooks. 4th state the composition of the nation neglecting the same distinctions and also that