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1o. A statement which explains the sense in which some word or phrase is employed is a definition.
A definition may select some one meaning out of several attached to a common word, or it may introduce some technical term to be used in a particular sense.
Some terms, such as space, straight, direction, etc., which express elementary ideas cannot be defined.
2o. Def.--A Theorem is the formal statement of some mathematical relation.
A theorem may be stated for the purpose of being subsequently proved, or it may be deduced from some previous course of reasoning.
In the former case it is called a Proposition, that is, something proposed, and consists of (a) the statement or enunciation of the theorem, and (b) the argument or proof. The purpose of the argument is to show that the truth of the theorem depends upon that of some preceding theorem whose truth has already been established or admitted.
Ex. "The sum of two odd numbers is an even number " is a theorem.
3o. A theorem so elementary as to be generally accepted as true without any formal proof, is an axiom.
Mathematical axioms are general or particular, that is, they apply to the whole science of mathematics, or have special applications to some department.
The principal general axioms are :-
i. The whole is equal to the sum of all its parts, and therefore greater than any one of its parts.
ii. Things equal to the same thing are equal to one
iii. If equals be added to equals the sums are equal.
v. If equals be added to or taken from unequals the results are unequal.
vi. If unequals be taken from equals the remainders are unequal.
vii. Equal multiples of equals are equal; so also equal submultiples of equals are equal.
The axioms which belong particularly to geometry will occur in the sequel.
4°. The statement of any theorem may be put into the hypothetical form, of which the type is-
If A is B then C is D.
The first part "if A is B" is called the hypothesis, and the second part "then C is D" is the conclusion.
Ex. The theorem "The product of two odd numbers is an odd number” can be arranged thus :
Hyp. If two numbers are each an odd number.
5°. The statement "If A is B then C is D" may be immediately put into the form
If C is not D then A is not B,
which is called the contrapositive of the former.
The truth of a theorem establishes the truth of its contra
positive, and vice versa, and hence if either is proved the other is proved also.
6°. Two theorems are converse to one another when the hypothesis and conclusion of the one are respectively the conclusion and hypothesis of the other.
Ex. If an animal is a horse it has four legs.
Converse. If an animal has four legs it is a horse.
As is readily seen from the foregoing example, the truth of a theorem does not necessarily establish the truth of its converse, and hence a theorem and its converse have in general to be proved separately. But on account of the peculiar relation existing between the two, a relation exists also between the modes of proof for the two. These are known as the direct and indirect modes of proof. And if any theorem which admits of a converse can be proved directly its converse can usually be proved indirectly. Examples will occur hereafter.
7°. Many geometric theorems are so connected with their converses that the truth of the theorems establishes that of the converses, and vice versa.
The necessary connection is expressed in the Rule of Identity, its statement being :—
If there is but one X and one Y, and if it is proved that X is Y, then it follows that Y is X.
Where X and Y stand for phrases such as may form the hypotheses or conclusions of theorems, and the " is " between them is to be variously interpreted as “equal to,” corresponds to," etc.
Ex. Of two sides of a triangle only one can be the greater, and of the two angles opposite these sides only one can be the greater. Then, if it is proved that the greater side is opposite the greater angle it follows that the greater angle is opposite the greater side.