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0011 0010 1010 1101 0001 0100 1011

Math for Elementary Teachers

Chapter 2Sets Whole Numbers, and

Numeration

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Sets as a Basis for Whole Numbers

Set a collection of objects 1. A verbal description2. A listing of the members separated by

commas or With braces {}3. Set-builder notation

Elements(members) objects in a set.

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Sets

Sets are denoted by capital letters A,B,C . indicates that an object is an element of

a set . indicates that an object is NOT an

element of a set . empty set (or null set) a set without

elements.

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Set Examples

Verbal the set of states that border the Pacific Ocean

Listing A:{Alaska, California, Hawaii, Oregon, Washington}

.Oregon A .New York A .The Set of all States bordering Iraq .

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More on Sets

Two sets are equal ( A=B) if and only if they have precisely the same elements Two sets, A and B, are equial if every elements

of A is in B, and vice versa If A does not equal B then A B

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Rules regarding Sets

1. The same element is not listed more than once within a set

2. the order of the elements in a set is immaterial.

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One-to-One Correspondence

Definition: A 1-1 correspondence between two sets A and B is a pairing of the elements of A with the elements of B so that each element of A corresponds to exactly one element 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 match.

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One-to-One Correspondence

Four possible 1-1 Equal sets are always

equivalent BUT equivalent sets

are not necessarily equal

{1,2}~{a,b} BUT {1,2} {a,b}.

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Subset of a Set: A B

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

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Subset examples:

Vermont is a subset of the set of all New England states

.

},,,,,{},,{ fedcbacba },,{},,{ dbacba

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Subset examples continued

If and B has an element that is not in A, we write and say that A is a proper subset of B

Thus , since and c is in the second set but not in the first.

BABA

},,{},{ cbaba },,{},{ cbaba

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Venn Diagrams

U = universe Disjoint Sets Sets A

and B have no elements in common

Sets {a,b,c} and {d,e,f} are disjoint

Sets {x,y} and {y,z} have y in common and are not disjoint.

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Union of Sets:

Definition: The union of two sets A and B, written is the set that consists of all elements belonging either to a or to b (or to both).

BA

BA

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Union of Sets:

. . The notion of set union is the basis for the

addition of whole numbers, but only when disjoint sets are used

2+3=5 .

},,,,{},,{},{ edcbaedcba =},,,{},,{},,{ qpnmpnmqnm =

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Intersection of Sets:

Definition: The intersection of sets A and B, written is the set of all elements common to sets A and B.

BAI

BAI

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Complement of a Set:

Definition: The complement of a set A,Written ,is the set of all elements in the universe, U, that are not in A.

A

A

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Difference of Sets: A-B

Definition: The set difference (or relative complement) a set B from set A, written A-B, is the set of all elements in A that are not in B.

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Section 2.2

Whole numbers and numeration

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Numbers and Numerals

The study of the set of whole numbers W={0,1,2,3,4} is the foundation of elementary school mathematics

A number is an idea, or an abstractions, that represents a quantity.

The symbols that we see, srite or touch when representing numbers are called numerals.

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Three uses of whole numbers

1. Cardinal number whole numbers used to describe how many elements are in a finite set

2. Ordinal numbers - concerned with order e.g. your team is in fourth place

3. Identification numbers used to name things credit card, telephone number, etc its a symbol for something.

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The symbol n(A) is used to represent the number of elements in a finite set A.

n({a,b,c})=3 n({a,b,c,,z})=26.

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Ordering Whole Numbers(1-1 correspondences)

Definition: Ordering Whole Numbers: Let a=n(A) and b=n(B)

then aa (b is greater than a) if A is equivalent to a proper subset of B.

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Problem: determine which is greater 3 or 8 in three different ways

1. Counting chant one, two, three, etc2. Set Method a set with three elements

can be matched with a proper subset of a set with eight elements 33.

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Problem: determine which is greater 3 or 8 in three different ways (cont)

3. Whole-Number Line since 3 is to the left of 8 on the number line, 3 is less than 8 and 8 is greater than 3.

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Numeration Systems

Tally numeration system single strokes, one for each object counted.

Improved with grouping.

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The Egyptian Numeration System

developed around 3400 B.C invovles grouping by ten.

=? 321.

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The Roman Numeration System

Developed between 500 B.C. and A.D. 100 The values are found by adding the values

of the various basic numerals MCVIII is 1000+100+5+1+1+1=1108 New elements

Subtractive principle Multiplicative principle.

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Subtractive system

Permits simplifications using combinations of basic numbers

IV take one from five instead of IIII

The value of the pair is the value of the larger less the value of the smaller.

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Multiplicative System

Utilizes a horizontal bar above a numeral to represent 1000 times the number

Then means 5 times 1000 or 5000 and is 1100 System still needs many more symbols than

current system and is cumbersome for doing arithmetic.

VXI

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The Babylonian Numeration System

Evolved between 3000 and 2000 B.C. Used only two numerals, one and ten

for numbers up to 59 system was simply additive Introduced the notion of place value symbols

have different values depending on the place they are written.

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Sections 2.3 The Hindu-Arabic System

1. Digits 0,1,2,3,4,5,6,7,8,9 10 digits can be used in combination to represent all possible numbers

2. Grouping by tens (decimal system) known as the base of the system Arabic is a base ten system

3. Place value (positional) Each of the various places in the number has its own value.

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Models for multi digit numbers

Bundles of Sticks each ten sticks bound together with a band

Base ten pieces (Dienes blocks) individual cubes grouped in tens.

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The Hindu-Arabic System

4. Additive and multiplicative The value of a Hindu-Arabic numeral is

found by multiplying each place value by its corresponding digit and then adding all of the resulting products.

Place values: thousand hundred ten oneDigits 6 5 2 3Numeral value 6x1000 + 5x100 + 2x10 + 3x1Numeral 6523.

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Observations about the naming procedure

1. The number 0,1,12 all have unique names2. The numbers 13,14, 19 are the teens3. The numbers 20,99 are combinations of earlier names

but reversed from the teens in that the tens place is named first e.g. 57 is fifty-seven

4. The number 100, 999 are combinations of hundreds and previous names e.g. 637 reads six hundred thirty-seven

5. In numerals containing more than three digits, groups of three digits are usually set off by commas e.g.123,456,789 .

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Learning

Three distinct ideas that children need to learn to understand the Hindu-Arabic numeration system .

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Base 5 operations

We can express numeration systems as base systems The number 18 in Hindu-Arabic can be stated

as 18ten 18 base ten To study a system with only five digits

(0,1,2,3,4) we would call that a base 5 systeme.g. base five 37five .