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DEFINITIONS AND SYMBOLS

The word "set" implies a collection or grouping of similar objects or symbols. The objects in a set have at lea& one characteristic in common, such as similarity of appearance or purpose. A set of tools would be an example of a group of objects not necessarily similar in appearance but similar in purpose. The objects or symbols in a set are called members or

ELEMENTS of the set.

The elements of a mathematical ret are usually symbols, ouch as numerals , lines, or points. For example, the integer6 greater than zero and less than 6 form a set, as follows:

    {1, 2, 3, 4}

Notice that braces are used to indicate sets. This is often done where the elements of the set are not too numerous, Since the elements of the set (2, 4, 6) are the same as the elements of (4, 2, 6}, there two seta are said to be equal. In other words, equality between sets has nothing to do with the order in which the elements are arranged. Further- more, repeated elements are not necessary. That is, the elements of (2, 2, 3, 4) are simply 2, 3, and 4. Therefore the sets (2, 3, 4) and (2, 2, 3,4) are equal. Practice problems:

1. Use the correct symbols to designate the set of odd positive integers greater than 0 and less than 10.
of names of days of the week which do not contain the letter "s".
3. List the elements of the
set of natural numbers greater than 15 and less than 20.
4. Suppose
that we have sets as follows:

A = (1, 2, 3)         C = (1, 2, 3, 4)

B = (1, 2, 2, 3)     D = (1, 1, 2, 3)

Which of these sets are equal?

Answers:

1. 1, 3, 5, 7, 9
2. {Monday, Friday} 
3. 16, 17, 18, and 19
4. A=B=D

SUBSETS

Since it is inconvenient to enumerate all of the elements of a set each time the set is mentioned, sets are often designated by a letter. For example, we could let S represent the set of all integers greater than 0 and less than 10. In symbols, this relationship could be stated as follows:

s = (1, 2, 3, 4, 5, 6, 7, 8,9)

Now suppose that we have another set, T, which comprises all positive even integers less than 10. This set is then defined as follows:

T - (2, 4, 6, 8)

Notice that every element of T is also an element of S. This establishes the SUBSET relationship; T is said to be a subset of 9.

POSITIVE INTEGERS

The mart fundamental ret of numbers is the set of positive integers. This ret comprises the counting numbers (natural numbers) and includes, as SUbsets, all of the sets of numbers which we have di8cueeed. The set of natural numbers has an outstanding characteristic: it is infinite. This means that the successive elements of the set continue to increase in size without limit, each number being larger by 1 than the number preceding it. Therefore there is no ‘largest" number; any number that we might choose as larger than all others could be increased to a larger number simply by adding 1 to it.

One way to represent the set of natural numbers symbolically would be as follows:

(1, 2, 3, 4, 5, 6, ...)

The three dots, called ellipsis, indicate that the pattern established by the numbers shown continues without limit. In other words, the next number in the set is understood to be 7, the next after that is 8, etc.

POINTS AND LINES

In addition to the many sets which can be formed with number symbols, we frequently find it necessary in mathematics to work with sets composed of points or lines.

A point is an idea, rather than a tangible object, just as a number is. The mark which is made on a piece of paper is merely a symbol representing the Point. In strict mathematical terms, a point has no dimensions (physical size) at all. Thus a Pencil dot is only a rough picture of a point, useful for indicating the location of the point but certainly not to be confused with the idea.

Now suppose that a large number of points are placed side by side to form a "string." Picturing this arrangement by drawing dots on paper, we would have a "dotted line." If more dots were placed between the dots already in the string, with the number of dots increasing until we could not see between them, we would have a rough picture of a line. Once again, it is important to emphasize that the picture is only a symbol which represents an ideal line. The ideal line would have length but no width or thickness.

The foregoing discussion leads to the conclusion that a line is actually a set of points. The number of elements in the set is infinite, since the line extends in both directions without limit.

The idea of arranging points together to form a line may be extended to the formation of planes (flat surfaces). A mathematical plane is determined by three points which do not lie on the same line. It is also determined by two intersecting lines.

Line Segments and Rays

When we draw a "line," label its end Points A and B, and call it ‘line AB," we really mean LINE SEGMENT AB. A line segment is a subset of the set of points comprising a line. men a line is considered to have a starting point but no stopping point (that is, it extends without limit in one direction), it is called a RAY. A ray is not a line segment, because it does not terminate at both ends; it may be appropriate to refer to a ray as a ‘half-line."

THE NUMBER LINE

As in the case of a line segment, a ray is a subset of the set of Points comprising a line. All three-lines, line segments, and rays-are subsets of the set of points comprising a plane, Among the many devices used for representing a set of numbers, one of the most useful is the number line. To illustrate the construction of a number line, let us place the elements of the set of natural numbers in one-to-one correspondence with points on a line. Since the natural numbers are equally spaced, we select points such that the distances between them are equal. The starting point is labeled 0, the next point is labeled 1, the next 2, etc., using the natural numbers in normal counting order. (See fig. l-3.) Such an arrangement is often referred to as a scale, a familiar example being the scale on a thermometer.

Thus far in our discussion, we have not mentioned any numbers other than integers. The number line is an ideal device for picturing the

Figure 1-3.-A number line.

relationship between integers and other numbers such as fractions and decimals. It is clear that many points, other than those representing integers, exist on the number line. Examples are the points representing the numbers l/2 (located halfway between 0 and 1) and 2.5 (located halfway between 2 and 3).

An interesting question arises, concerning the "in-between" points on the number line:

How many points (numbers) exist between any two integers? To answer this question, suppose that we first locate the point halfway between 0 and 1, which corresponds to the number l/2. Then let us locate the point halfway between 0 and l/2, which corresponds to the number l/4. The result of the next such halving operation could be l/8, the next l/16, etc. If we need more space to continue our halving operations on the number line, we can enlarge our "picture" and then continue. 

It soon becomes apparent that the halving process could continue indefinitely; that is, without limit. In other words, the number of points between 0 and 1 is infinite. The same is true of any other interval on the number line. Thus, between any two integers there is an infinite set of numbers other than integers. If this seems physically impossible, considering that even the sharpest pencil point has some width, remember that we are working with ideal points, which have no physical dimensions whatsoever. Although it is beyond the scope of this course to discuss such topics as orders of infinity, it is interesting to note that the set of integers contains many subsets which are themselves infinite. Not only are the many subsets of numbers other than integers infinite, but also such subsets as the set of all odd integers and the set of all even integers. By intuition we see that these two subsets are infinite, as follows: If we select a particular odd or even integer which we think is the largest possible, a larger one can be formed immediately by merely adding 2.

Perhaps the most practical use for the number line is in explaining the meaning of negative numbers. Negative numbers are discussed in detail in chapter 3 of this course.




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