CHAPTER 1

SERIES AND MATHEMATICAL INDUCTION
Not And Or Implies Equals

1. In the following pages we will find it convenient to use
several symbols of logic, - for not, ~ for and, — for or, D for
implies.

Thus “2 —= 3" stands for ‘2 is not equal to 3”’; “2-> 3”
stands for ““2 is not greater than 3""; “5 > 2 ~ 3 < 7" stands
for “‘S is greater than 2 and 3 is less than 7”; “x and y are
numbers ~xy = 0D «x = 0—y = 0" may be read “x and y
are numbers and xy = 0 implies that x = 0 or y = 0,” that is,
“if the product of two numbers is zero, one of them must
be zero.”

There is a fifth idea of logic, the symbol for which we will
find it convenient to use. Sometimes of two statements each
implies the other. Thus, if xisa number,3x +2 = 8§ D x = 2
and x = 2D 3x + 2 = 8. In such a case we say that the
statements are equal, or equivalent. For this kind of equality
we use the same sign as for equality of numbers. Thus we
write

Gr+2=28) = (x=2).

2. The Principle of Mathematical Induction in perhaps its
simplest form may be stated as follows:

If the first of a series (or progression) of statements is tru and
the truth of any one implies the truth of the next one, if there is
a next one, then the statements are all tru.

3. The truth of this principle seems to be self evident.
Many persons, however, when they first meet the principle,
do not readily grasp the meaning of the words in which it is
stated and therfor do not grant its truth. It is hoped that

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