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Sets, Whole Numbers,and Numeration

The Mayan Numeration System

CHAPTER

2

The Maya people lived mainly in southeastern Mexico,including the Yucatan Peninsula, and in much ofnorthwestern Central America, including Guatemala

and parts of Honduras and El Salvador. Earliest archaeo-logical evidence of the Maya civilization dates to 9000B.C.E., with the principal epochs of the Maya cultural development occurring between 2000 B.C.E. and C.E. 1700.

Knowledge of arithmetic, calendrical, and astronomicalmatters was more highly developed by the ancient Mayathan by any other New World peoples. Their numerationsystem was simple, yet sophisticated. Their system uti-lized three basic numerals: a dot, , to represent 1; a hori-zontal bar, , to represent 5; and a conch shell, , torepresent 0. They used these three symbols, in combina-tion, to represent the numbers 0 through 19.

The sun, and hence the solar calendar, was very impor-tant to the Maya. They calculated that a year consisted of365.2420 days. (Present calculations measure our year as365.2422 days long.) Since 360 had convenient factorsand was close to 365 days in their year and 400 in their numeration system, they made a place value systemwhere the column values from right to left were 1, 20, 20 18 ( 360), 202 18 ( 7200), 203 18 ( 144,000), andso on. Interestingly, the Maya could record all the days oftheir history simply by using the place values through144,000. The Maya were also able to use larger numbers.One Mayan hieroglyphic text recorded a number equiva-lent to 1,841,641,600.

Finally, the Maya, famous for their hieroglyphic writing,also used the 20 ideograms pictured here, called head variants,to represent the numbers 019.

43

FOCUS ON

For numbers greater than 19, they initially used a base20 system. That is, they grouped in twenties and displayedtheir numerals vertically. Three Mayan numerals areshown together with their values in our system and theplace values initially used by the Mayans. The Mayan numeration system is studied in this chapter

along with other ancient numeration systems.

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Often there are problems where, although it is not necessary to draw an ac-tual picture to represent the problem situation, a diagram that represents theessence of the problem is useful. For example, if we wish to determine thenumber of times two heads can turn up when we toss two coins, we could lit-erally draw pictures of all possible arrangements of two coins turning upheads or tails. However, in practice, a simple tree diagram is used like theone shown next.

44

STRATEGY 7Draw a Diagram

Problem-Solving Strategies1. Guess and Test

2. Draw a Picture

3. Use a Variable

4. Look for a Pattern

5. Make a List

6. Solve a Simpler Problem

7. Draw a Diagram

INITIAL PROBLEM

CLUES

A survey was taken of 150 college freshmen. Forty of them were majoring inmathematics, 30 of them were majoring in English, 20 were majoring in sci-ence, 7 had a double major of mathematics and English, and none had a dou-ble (or triple) major with science. How many students had majors other thanmathematics, English, or science?

The Draw a Diagram strategy may be appropriate when

The problem involves sets, ratios, or probabilities. An actual picture can be drawn, but a diagram is more efficient. Relationships among quantities are represented.

A solution of this Initial Problem is on page 82.

This diagram shows that there is one way to obtain two heads out of fourpossible outcomes. Another type of diagram is helpful in solving the nextproblem.

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Section 2.1 Sets As a Basis for Whole Numbers 45

I N T R O D U C T I O N

Key Concepts from NCTM Curriculum Focal Points

PREKINDERGARTEN: Developing an understanding of whole numbers, includ-ing concepts of correspondence, counting, cardinality, and comparison.

KINDERGARTEN: Representing, comparing, and ordering whole numbers andjoining and separating sets. Ordering objects by measurable attributes.

GRADE 1: Developing an understanding of whole number relationships, includ-ing grouping in tens and ones.

GRADE 2: Developing an understanding of the base-ten numeration system andplace-value concepts.

GRADE 6: Writing, interpreting, and using mathematical expressions and equations.

uch of elementary school mathematics is devoted to the study of numbers.Children first learn to count using the natural numbers or counting numbers1, 2, 3, . . . (the ellipsis, or three periods, means and so on). This chapter de-

velops the ideas that lead to the concepts central to the system of whole numbers 0, 1,2, 3, . . . (the counting numbers together with zero) and the symbols that are used to rep-resent them. First, the notion of a one-to-one correspondence between two sets is shownto be the idea central to the formation of the concept of number. Then operations on setsare discussed. These operations form the foundation of addition, subtraction, multipli-cation, and division of whole numbers. Finally, the HinduArabic numeration system,our system of symbols that represent numbers, is presented after its various attributesare introduced by considering other ancient numeration systems.

M

2.1 SETS AS A BASIS FOR WHOLE NUMBERS

After forming a group of students, use a diagramlike the one at the right to place the names of eachmember of your group in the appropriate region. Allmembers of the group will fit somewhere in the rec-tangle. Discuss the attributes of a person whosename is in the shaded region. Discuss the attributesof a person whose name is not in any of the circles.

STARTING POINT

SetsA collection of objects is called a set and the objects are called elements or membersof the set. Sets can be defined in three common ways: (1) a verbal description, (2) alisting of the members separated by commas, with braces ({ and }) used to en-close the list of elements, and (3) set-builder notation. For example, the verbal de-scription the set of all states in the United States that border the Pacific Ocean canbe represented in the other two ways as follows:

Algebraic ReasoningWhen solving algebraicequations, we are looking for all values/numbers thatmake an equation true. This set of numbers is called thesolution set.

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1. Listing: {Alaska, California, Hawaii, Oregon, Washington}.

2. Set-builder: {x x is a U.S. state that borders the Pacific Ocean}. (This set-buildernotation is read: The set of all x such that x is a U.S. state that borders the PacificOcean.)

Sets are usually denoted by capital letters such as A, B, C, and so on. The symbols and are used to indicate that an object is or is not an element of a set, re-spectively. For example, if S represents the set of all U.S. states bordering the Pacific,then Alaska S and Michigan S. The set without elements is called the empty set(or null set) and is denoted by { } or the symbol . The set of all U.S. states border-ing Antarctica is the empty set.

Two sets A and B are equal, written A B, if and only if they have precisely thesame elements. Thus {x x is a state that borders Lake Michigan} {Illinois, Indi-ana, Michigan, Wisconsin}. Notice that two sets, A and B, are equal if every elementof A is in B, and vice versa. If A does not equal B, we write A B.

There are two inherent rules regarding sets: (1) The same element is not listedmore than once within a set, and (2) the order of the elements in a set is immaterial.Thus, by rule 1, the set {a, a, b} would be written as {a, b} and by rule 2. {a, b} {b, a}, {x, y, z} {y, z, x}, and so on.

The concept of a 1-1 correspondence, read one-to-one correspondence, isneeded to formalize the meaning of a whole number.

46 Chapter 2 Sets, Whole Numbers, and Numeration

One-to-One Correspondence

A 1-1 correspondence between two sets A and B is a pairing of the elements of Awith the elements of B so that each element of A corresponds to exactly one ele-ment of B, and vice versa. If there is a 1-1 correspondence between sets A and B,we write A B and say that A and B are equivalent or matching sets.

D E F I N I T I O ND E F I N I T I O N

Figure 2.1 shows two 1-1 correspondences between two sets, A and B.There are four other possible 1-1 correspondences between A and B. Notice that

equal sets are always equivalent, since each element can be matched with itself, but that equivalent sets are not necessarily equal. For example, {1, 2} {a, b}, but {1, 2} {a, b}. The two sets A {a, b} and B {a, b, c} are not equivalent. How-ever, they do satisfy the relationship defined next.

Subset of a Set: A B

Set A is said to be a subset of B, written A B, if and only if every element of A isalso an element of B.

D E F I N I T I O ND E F I N I T I O N

The set consisting of New Hampshire is a subset of the set of all New Englandstates and {a, b, c} {a, b, c, d, e, f}. Since every element in a set A is in A, A Afor all sets A. Also, {a, b, c} {a, b, d} because c is in the set {a, b, c} but not in the set {a, b, d}. Using similar reasoning, you can argue that A for any set A since itis impossible to find an element in that is not in A.

Figure 2.1

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If A B and B has an element that is not in A, we write A B and say that A is aproper subset of B. Thus {a, b} {a, b, c}, since {a, b} {a, b, c} and c is in thesecond set but not in the first.

Circles or other closed curves are used in Venn diagrams (named after the En-glish logician John Venn) to illustrate relationships between sets. These circles areusually pictured within a rectangle, U, where the rectangle represents the universalset or universe, the set comprised of all elements being considered in a particular dis-cussion. Figure 2.2 displays sets A and B inside a universal set U.

Figure 2.2

Reflection from ResearchWhile many students will agreethat two infinite sets such as the set of counting numbers andthe set of even numbers areequivalent, the same studentswill argue that the set ofcounting numbers is larger dueto its inclusion of the oddnumbers (Wheeler, 1987).

Section 2.1 Sets As a Basis for Whole Numbers 47

Set A is comprised of everything inside circle A, and set B is comprised of everythinginside circle B, including set A. Hence A is a proper subset of B since x B, but x A. The idea of proper subset will be used later to help establish the meaning of the concept less than for whole numbers.

Check for Understanding: Exercise/Problem Set A #19

Finite and Infinite SetsThere are two broad categories of sets: finite and infinite. Informally, a set is finite ifit is empty or can have its elements listed (where the list eventually ends), and a set isinfinite if it goes on without end. A little more formally, a set is finite if (1) it is emptyor (2) it can be put into a 1-1 correspondence with a set of the form {1, 2, 3, . . . , n},where n is a counting number. On the other hand, a set is infinite if it is not finite.

Determine whether the following sets are finite or infinite.

a. {a, b, c}b. {1, 2, 3, . . .}c. {2, 4, 6, . . . , 20}

SOLUTIONa. {a, b, c} is finite since it can be matched with the set {1, 2, 3}.b. {1, 2, 3, . . .} is an infinite set.c. {2, 4, 6, . . . , 20} is a finite set since it can be matched with the set {1, 2, 3, . . . ,

10}. (Here, the ellipsis means to continue the pattern until the last element isreached.)

NOTE: The small solid square ( ) is used to mark the end of an example or mathe-matical argument.

Example 2.1

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An interesting property of every infinite set is that it can be matched with a propersubset of itself. For example, consider the following 1-1 correspondence:

48 Chapter 2 Sets, Whole Numbers, and Numeration

Algebraic ReasoningDue to the nature of infinite sets,a variable is commonly used toillustrate matching elements intwo sets when showing 1-1correspondences.

Figure 2.3

There are many ways to construct a new set from two or more sets. The followingoperations on sets will be very useful in clarifying our understanding of whole num-bers and their operations.

Union of Sets: A B

The union of two sets A and B, written A B, is the set that consists of all ele-ments belonging either to A or to B (or to both).

D E F I N I T I O ND E F I N I T I O N

Informally, A B is formed by putting all the elements of A and B together. Thenext example illustrates this definition.

Find the union of the given pairs of sets.

a. {a, b} {c, d, e}b. {1, 2, 3, 4, 5} c. {m, n, q} {m, n, p}

SOLUTIONa. {a, b} {c, d, e} {a, b, c, d, e}b. {1, 2, 3, 4, 5} {1, 2, 3, 4, 5}c. {m, n, q} {m, n, p} {m, n, p, q}

Example 2.2

Note that B A and that each element in A is paired with exactly one element inB, and vice versa. Notice that matching n with 2n indicates that we never run out ofelements from B to match with the elements from set A. Thus an alternative definitionis that a set is infinite if it is equivalent to a proper subset of itself. In this case, a set isfinite if it is not infinite.

Check for Understanding: Exercise/Problem Set A #1011

Operations on SetsTwo sets A and B that have no elements in common are called disjoint sets. The sets{a, b, c} and {d, e, f } are disjoint (Figure 2.3), whereas {x, y} and {y, z} are not dis-joint, since y is an element in both sets.

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Notice that although m is a member of both sets in Example 2.2(c), it is listed onlyonce in the union of the two sets. The union of sets A and B is displayed in a Venndiagram by shading the portion of the diagram that represents A B (Figure 2.4).

Section 2.1 Sets As a Basis for Whole Numbers 49

Figure 2.4 Shaded region is A B.

Figure 2.5 Shaded region is A B.

The notion of set union is the basis for the addition of whole numbers, but only whendisjoint sets are used. Notice how the sets in Example 2.2(a) can be used to show that 2 3 5.

Another useful set operation is the intersection of sets.

Intersection of Sets: A B

The intersection of sets A and B, written A B, is the set of all elements commonto sets A and B.

D E F I N I T I O ND E F I N I T I O N

Thus A B is the set of elements shared by A and B. Example 2.3 illustrates thisdefinition.

Find the intersection of the given pairs of sets.

a. {a, b, c} {b, d, f }b. {a, b, c} {a, b, c}c. {a, b} {c, d}

SOLUTIONa. {a, b, c} {b, d, f } {b} since b is the only element in both sets.b. {a, b, c} {a, b, c} {a, b, c} since a, b, c are in both sets.c. {a, b} {c, d} since there are no elements common to the given two sets.

Figure 2.5 displays A B. Observe that two sets are disjoint if and only if theirintersection is the empty set. Figure 2.3 shows a Venn diagram of two sets whoseintersection is the empty set.

Example 2.3

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In many situations, instead of considering elements of a set A, it is more productive toconsider all elements in the universal set other than those in A. This set is defined next.

50 Chapter 2 Sets, Whole Numbers, and Numeration

The set is shaded in Figure 2.6.A

I...

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