Miscellaneous Mathematical Definitions

Article. Miscellaneous Definitions

This article is the repository of all definitions that don't seem to t
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Table of Contents

1. Various Number Systems
•The Natural Numbers •The Integers •The Rational Numbers
The Real Numbers
2. Intervals
•Various Intervals •Partitioning an Interval
3. Functions
•Bounded Functions

1. Various Number Systems

In this section we give a brief discussion of the various number systems.

•The Natural Numbers
Definition 1.1. The set of natural numbers, denoted by N, is defined
to be

The natural numbers are also known as the set of all positive integers .

•The Integers
Definition 1.2. The set of all integers, denoted by Z, is defined to
be

Thus, one might think of the integers, Z, as consisting of three distinct
parts. The positive integers, N, the set of all natural numbers,
the negative integers , which could be though of as −N (the negation
of every number in N), and 0. Symbolically,

That's a nice formula .

•The Rational Numbers
Definition 1.3. A number, r, is a rational number if it is the ratio
of two integers , i.e. a rational number r has the form

The set of all rational numbers is denoted Q.

Q = { r|r is a rational number }.

•The Real Numbers

Definition 1.4. An irrational number is a number that cannot be
expressed as the ratio of two integers. The set of all real numbers,
denoted R, is defined by

R = { x|x is either a rational or irrational number }.

In interval notation, R is written as

Within the context of Analytic Geometry, R is also known as the
x-axis.

2. Intervals

•Various Intervals
Definition 2.1. An interval I is called an open interval if I does not
contain its endpoints.

Examples. There are two types. Intervals of finite length, and intervals
of infinite  length.

Intervals of Finite Length. Let a < b be real numbers. Then

is an open interval.
Intervals of infinite  Length. Let a ∈R. Then each of the following are
open intervals.

Another important type of interval is the closed interval.

Definition 2.2. An interval I is called an closed interval if the endpoints
of I belong to the interval.

Examples. A general example is, for a≤b,

A particular example is [ 0, 1 ].

Symmetrical Intervals. An interval I of number is said to be symmetrical
about the origin provided

Examples of symmetrical intervals are (−1, 1), [−3, 3], . The
intervals are not symmetrical about the origin. (−2, 3), [1, 2],

•Partitioning an Interval
Definition 2.3. Let [ a, b ] be a closed interval. A partition, P, of
[a, b ] is any finite subset of [ a, b ] that contains the numbers a and b.
Or, more symbolically, a finite set

is a partition of [ a, b ] provided and . (Here,
.)

Definition Notes. The labeling used in (1) is the standard way of
symbolically writing the elements of a partition.

•The elements of a partition are called partition points or nodes.
•When we write the elements of a partition it is customary to
have them labeled such that

•With the convention established in the previous point, and the
fact that a, b ∈[ a, b ], it follows that x₀ = a and x_n = b. Thus,

A visualization of a partition can be seen from the next diagram.

Partitioning SchemeYou can see from the chart above how the nodes partition, or subdivide
the interval into pieces.

•If P is a partition as defined in (1), then the P also subdivides
the interval into subintervals. These subintervals, for example, are
used as a basis for the construction of the De finite Integral. The nodes
of the partition P are used as endpoints of these subintervals. Below is
a listing of the subinterval as well as the usual scheme for numbering
them.

•From the above listing of the interval, it is clear the the partition,
as given in equation (1), sub divides the interval into n subintervals.
This is the significance of the natural number n in (1). You'll note
that it takes n+1 nodes to subdivide the interval [ a, b ] into n parts.

•Finally, we note that the index variable , i, is used to manipulate
the various elements of a partition. For i = 1, 2, 3, . . . , n, the i^th node
is xi and the i^th subinterval is . The length of the i^th
subinterval, , is typically denoted by the symbol
The calculated length of the i^th subinterval is given by the formula

which is nothing more than the value of the upper endpoint, x_i, of
the i^th subinterval minus the value of the lower endpoint, x_i−1 of this
interval.

Example. Here is a simple example to illustrate. Let the interval
[ a, b ] be the [ 0,1 ]. The set

is a partition of the interval [ 0, 1 ] since and . A
natural question to ask is, "Where is all the elaborate label system?"
The labeling system is there , you just have to use it.

Let's index the nodes.

(x₀ is always the left-hand endpoint.)

(The last node is always the right-hand endpoint.)

We can see now that n = 4. (The natural number n is the index
number of the right-hand endpoint, or more simply, n is one-less the
number of nodes in the partition-we have 5 nodes, so n must be 4.)

Thus, the partition P in equation (2) subdivides the interval [ 0, 1 ]
into n = 4 subintervals.

First Sub-interval.

Second Sub-interval.

Third Sub-interval.

Fourth Sub-interval.

Lastly, the length of the 3^rd subinterval is obtained by taking the
general formula

and putting x = 3,

Of course, in this simple example, we could have computed the length
of the 3^rd subinterval by taking this interval , as computed
above, and calculated its length The first method is
useful in abstract discussions, the latter is used for specific examples.

3. Functions

•Bounded Functions
Definition 3.1. Let y = f(x) be a real-valued function having domain
Dom(f)R. Let ADom(f). We say that the function f is
bounded over the set A, if there is some number M >0 such that

In this case, we say that M is a bound for f over A and that f is
bounded by M over the set A.

Definition Notes. Algebraically , the absolute inequality in (1) is equivalent
to

•In terms of geometry , if we were to draw the graph of f over
the set A, and draw the horizontal lines y = −M and y = M, then
the graph of f over the set A does not go below the horizontal line
y = −M and does not go above the horizontal line y = M.

Or, said more simply, a function f is bounded over the set A if
the graph of f lies between two horizontal lines.

A function that is not bounded over a set A is said to be unbounded
over that set.

It is convenient to create two related notions. bounded below
and bounded above. A function f is bounded below over A if there
exists a number m such that f(x)≥m, for all x ∈A. A function f is
bounded above over A if there exists a number M such that f(x)≤M,
for all x ∈A.

The definition of boundedness can be rewritten. f is bounded
over the set A, if f is both bounded above and bounded below over the
set A.

Examples of Boundedness. The function f(x) = x^2 is bounded over
the interval [ 0, 1 ]. Indeed, for all x ∈[ 0, 1 ]. We say, in this
case, that f is bounded above by 1, since f(x)≤1 for all x ∈[ 0, 1 ],
and bounded below be 0, since f(x)≥0 for all x ∈[ 0, 1 ]. The same
function f(x) = x^2 is unbounded over the interval , however,

over that interval, f is bounded below by 0 - it just doesn't have an
upper bound over that interval.