A Theorem is the formal statement of a property that may be demonstrated from known propositions. These propositions may themselves be theorems or axioms. A theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom. Thus, in the typical theorem,
the hypothesis is that X is Y , and the conclusion is that Z is W.
Converse Theorems.—Two theorems are said to be converse, each of the other, when the hypothesis of either is the conclusion of the other. Thus the converse of the theorem (i.) is—
From the two theorems (i.) and (ii.) we may infer two others, called their contrapositives. Thus the contrapositive
| of (i.) is, If Z is not W, then X is not Y ; | (iii.) | |
| of (ii.) is, If X is not Y , then Z is not W. | (iv.) |
The theorem (iv.) is called the obverse of (i.), and (iii.) the obverse of (ii.).
A Problem is a proposition in which something is proposed to be done, such as a line to be drawn, or a figure to be constructed, under some given conditions.
The Solution of a problem is the method of construction which accomplishes the required end.