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The IntegersOne of the most important sets in all of mathematics is the set Z of integers:
Z = {...,3,2,1,0,1,2,3...}.
In spite of centuries of effort by the greatest mathematical minds, there are many simple problems concerning Z which cannot be settled at the present time. For example, consider the problem of Fermat which was mentioned in historical perspectives, concerning the nonexistence of nonzero solutions of the equation It is possible to construct the integers in terms of sets and to verify all the basic properties from the properties of sets. However, this is a rather drawnout process, and would take us far afield from our central area of interest, so we will resist the temptation to provide a complete development of the integers. The Basic Properties of ZIn the following discussion, we will present five basic sets of properties of the integers. Mathematicians have distilled these properties of the integers during centuries of investigations. The properties which we will assume have two convenient features: They agree with our preconceived notions about the integers and they form a natural point of departure for the development of less elementary properties. On the set of integers, there are defined two binary operations, one called addition (denoted +) and one called multiplication (denoted ·). Our first three sets of properties of the integers are just some familiar properties of + and · .
I. Properties of AdditionA1. Associativity: If
A2. Commutativity: If
A3. Identity: If
If
A4. Inverses: If
II. Properties of Multiplication
M1. Associativity: If
M2. Commutativity: If
M3. Identity: If
III. Distributive Law
Addition and multiplication in Z are related by the distributive
law which asserts that if
a · (b + c) = a · b + a · c.
At this point, let us cite some rules for calculating with integers. These rules are familiar from high school algebra and can easily be deduced from I, II, and III. Let
0 = 0, 0 · x = 0,
(2)
(x + y) = x + (y),
(3)
(x) = x,
(4)
(1) · x = x,
(5)
(x) · (y) = x · y,
(6)
x · (y) = (x) · y = (x · y).
By way of illustration, let us prove (3). By A4, y is the unique integer which has the property The elements of the set
P = {1,2,3,...} are called positive integers. The set P of positive integers is a subset of Z. Our next set of properties concerns the positive integers.
IV. Properties of Positive Integers
Exactly one of the following is true:
P0. Let The set is called the set of natural numbers. Our last basic property of Z is the principle of mathematical induction.
V. Principle of Mathematical InductionLet S be a nonempty subset of the set of positive integers P. Assume that In more formal developments of the integers, the principle of mathematical induction is the basic tool available to prove theorems and, in fact, all the preceding basic properties may be deduced by using the principle of mathematical induction and a set of axioms for N called Peano's axioms.
The principle of mathematical induction can be used to prove theorems
as follows:Suppose that for each positive integer n, we have a given
proposition P_{n}. Suppose further that we are able to show that P_{1} is true and that whenever P_{n} is true, P_{n+1} is true. Then P_{n} is true for every positive integer n. Indeed if
For each positive integer n there is a proposition P_{n}. Suppose also that
(a) P_{1} is true and that
(b) whenever P_{m} is true for all V'. Variant of Mathematical Induction PrincipleSuppose that for each positive integer n there is a proposition P_{n}. Suppose also that Let us illustrate the use of the principle of mathematical induction by means of a number of examples. Example 1: Let us prove that in n is a positive integer, then
1 + 2 + ... + n = n(n + 1)/2
Let P_{n} denote this proposition. It is clear that P_{1} is true since
1 + 2 + ... + n = n(n + 1)/2.
But then by adding n + 1 to both sides of the equation, we derive that
1 + 2 + ... + n + (n + 1) = n(n + 1)/2 + (n + 1) = (n + 1)(n + 2)/2
and thus P_{n+1} is true. Thus, by the principle of mathematical induction, P_{n} is true for all Example 2: Let S be a set containing n elements. Then S contains 2^{n} subsets. Denote this proposition by P_{n}. It is clear that P_{1} since if T_{1}{a_{n+1}},...,T_{2n
}{a_{n+1}}.
We have exhibited
Example 3: Definition by induction. It is possible to use
mathematical induction to define various mathematical concepts. The basic idea of this process of "definition by induction" is as follows: Suppose that it is required to define a function a^{0} = 1,
a^{1} = a,
a^{n+1} = a^{n} · a.
By principle of mathematical induction, a^{n} is defined for n a positive integer or 0.
Example 4: Let (x·y)^{m} = x^{m} · y^{m},
(8)
x^{m} · x^{n} = x^{m+n},
(9)
(x^{m})^{n} = x^{m·n}. Feel free to complete the proof of these yourself. OrderLet a and b be integers. We say that a is greater than b, denoted a>b, if ab is a positive number. Thus, in particular, if
Exactly on of the following holds:
O1. Exactly on of the following holds: For, by P0, exactly one of the following holds:
In the first case
O2. If A simple application of O2 and O3 shows that there are no integers between 0 and 1. More precisely, we have If b is a positive integer. Then b is greater than or equal to 1 Proposition 1: Suppose that b is a positive integer. Then b is greater than or equal to 1.
Proof: Let us proceed by induction. The assertion is clearly true for
b + 1 > 1 + 1, 1 + 1 > 1. Therefore by O2
b + 1 > 1, which implies our assertion for b + 1. Therefore, the proposition is proved.
Let x and y be integers. If We will require one additional basic property of inequalities:
If x < y and z > 0 then xz < yz. O4a. If x < y and z > 0 then xz < yz. O4b. If x < y and z < 0 then xz > yz.
For if Another easy consequence of Proposition 1 is given by
Let a and b positive integers. Then there exists a positive integer n such that
Corollary 3: Let a and b positive integers. Then there exists a positive integer n such that
Proof: Set
nb > a + 0 = a.
A very important property of Z is the fact that if a and b are nonzero integers, then a · b is nonzero. The simplest way to see this is to divide the reasoning up into four cases:
Theorem 4: Let a and b be nonzero integers. Then If S is a subset of Z, then an integer s is said to be the least(or smallest )element of S if Proposition 5: Let S be a finite, nonempty subset of Z. Then S has a largest and smallest element. A somewhat more difficult result to prove is the wellordering principle: Theorem 6: Let S be a nonempty (finite or infinite) subset of N. Then S contains a smallest element. Proof: The following argument makes a rather interesting use of the principle of mathematical induction. Let
The Division AlgorithmLet a be a nonnegative integer and let b be a positive integer. In elementary arithmetic, we learned how to "divide" a by b, thereby deriving a quotient q plus a remainder of the form r/b, where (10)
0 < r/b < 1. This process of division can be expressed by the equation (11)
a/b = q + r/b. It is possible to formulate the process of division, as described by (10) and (11), solely in terms of integers.
Theorem 7(Division Algorithm): Let a be a nonnegative integer and let b be a positive integer. Then there exist natural numbers q and t such that
a = bq + r.
Proof: Of all the multiples
0 · b, 1 · b, 2 · b, 3 · b, ...
of b, there are only finitely many less than or equal to a. Indeed, Corollary 3 asserts that there exists a positive number n such that
r = a  qb < (q + 1) · b  qb = b
since The division algorithm, although it appears to be a rather superficial feature of the integers, is actually one of the most important results about Z. In other sections, we will see that the division algorithm can be applied in rather ingenious ways to yield arithmetic facts about the integers.

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