434. The Conjunctive Combination of Alternative Terms.—The first law of distribution gives the general rule for the conjunctive combination of alternatives. But with a view to such combination special attention may be called (i) to the second law of distribution, namely, (A or B) (A or C) = A or BC ; and (ii) to the equivalence (A or B) (AC or D) = AC or AD or BD, which may be established as follows: By the first law of distribution (A or B) (AC or D) is equivalent to AAC or ABC or AD or BD ; but by the law of simplicity AAC = AC, and by the law of absorption AC or ABC = AC ; hence our original term is equivalent to AC or AD or BD, which was to be proved.
From the above equivalences we obtain the two following practical rules which are of great assistance in simplifying the process of conjunctively combining alternatives:
(1) If two alternatives which are to be conjunctively combined have an alternant in common, this alternant may be at once written down as one alternant of the result, and we need not go through the form of combining it with any of the remaining alternants of either alternative;
(2) If two alternatives are to be conjunctively combined and an alternant of one is a subdivision of an alternant of the other, then the former alternant may be at once written down as one alternant of the result, and we need not go through the form of combining it with the remaining alternants of the other alternative.[483]
[483] These rules are equivalent to Boole’s second Method of Abbreviation (Laws of Thought, p. 131).
EXERCISES.
435. Simplify the following terms: (i) AD or acD ; (ii) Ad or Ae or aB or aC or aE or bC or bd or bE or be or cd or ce. [K.]
(i) By rule (1) in section [433], AD or acD is equivalent to (A or ac) D ; and this by rule (9) is equivalent to (A or c) D ; which again by rule (1) is equivalent to AD or cD.[484]
(ii) The dual term bE or be may be reduced to b, and hence Ad or Ae or aB or aC or aE or bC or bd or bE or be or cd or ce = Ad or Ae or aB or aC or aE or b or bC or bd or cd or ce. By section 433, rule (7), we may now omit all alternants in which b occurs as a determinant, and by rule (9), B may be omitted wherever it occurs as a determinant; accordingly our term is reduced to Ad or Ae or a or aC or aE or b or cd or ce. Since a is now an alternant, a further application of the same rules leaves us with a or b or cd or ce or d or e ; and this is immediately reducible to a or b or d or e.
[484] We might also proceed as follows: AD or acD = AD or AcD or acD [by rule (7)] = AD or cD [by rule (5)].
436. Shew that BC or bD or CD is equivalent to BC or bD. [K.]