470. Establish the following: Where A is present, either B and C are both present or C is present D being absent or C is present F being absent or H is present; therefore, where C is absent, A cannot be present H being absent. [K.]
471. Given that Whatever is PQ or AP is bCD or abdE or aBCdE or Abcd, shew that (1) All abP is CD or dE or q ; (2) All DP is bC or aq ; (3) Whatever is B or Cd or cD is a or p ; (4) All B is C or p or aq ; (5) All AB is p ; (6) If ae is c or d it is p or q ; (7) If BP is c or D or e it is aq. [K.]
472. Bring the following propositions to the form Everything is either X1 or X2 … or Xn:—
Whatever is Ac or ab or aC is bdf or deF ;
Nothing that is A and at the same time either B or C is D or dE. [K.]
473. Shew that the results in section [447] follow from those in section [446] by the rules of contradiction and contraposition. [K.]
CHAPTER IV.
THE COMBINATION OF COMPLEX PROPOSITIONS.
474. The Problem of combining Complex Propositions.—Two or more complex propositions given in simple combination, either conjunctive or alternative, constitute a compound proposition. Hence the problem of dealing with a combination of complex propositions so as to obtain from them a single equivalent complex proposition, which is the problem to be considered in the present chapter, is identical with that of passing from a compound proposition to an equivalent complex proposition; and it is, therefore, the converse of the problem which was partially discussed in sections [446], 447. The latter problem, namely, that of passing from a complex to an equivalent compound proposition, will be further discussed in [chapter 6].
475. The Conjunctive Combination of Universal Affirmatives.—We may here distinguish two cases according as the propositions to be combined have or have not the same subject.
(1) Universal affirmatives having the same subject.