Math for Elementary Teachers
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Chapter 2
Sets Whole Numbers, and
Numeration
Sets as a Basis for Whole Numbers
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Set – a collection of objects
1. A verbal description
2. A listing of the members separated by
commas or With braces {}
3. Set-builder notation
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Elements(members) – objects in a set.
Sets
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• 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
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• 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
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• Two sets are equal ( A=B) if and only if
they have precisely the same elements
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– 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
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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
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• 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 11 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
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• 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
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• Definition: Set A is said to be a subset of B,
 if and only if every element
written A B,
of A is also an element of B.
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Subset examples:
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• Vermont is a subset of the set of all New
England states
• {a, b, c}  {a, b, c, d , e, f }
• .{a, b, c} {a, b, d }
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Subset examples continued
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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 a proper
subset of B
• Thus {a, b}  {a, b, c}, since {a, b}  {a, b, c} and c is
in the second set but not in the first.
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Venn Diagrams
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• 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: A  B
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• Definition: The union of two sets A and B,
written A  B is the set that consists of all
elements belonging either to a or to b (or to
both).
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Union of Sets:
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{.a, b} {c, d , e}  {a, b, c, d , e}
{.m, n, q} {m, n, p}  {m, n, p, q}
The notion of set union is the basis for the
addition of whole numbers, but only when
disjoint sets are used
• 2+3=5 .
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Intersection of Sets: A  B
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• Definition: The intersection of sets A and
B, written A  B is the set of all elements
common to sets A and B.
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Complement of a Set: A
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• Definition: The complement of a set A,
Written A ,is the set of all elements in the
universe, U, that are not in A.
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Difference of Sets: A-B
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• 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
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Whole numbers and numeration
Numbers and Numerals
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• 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
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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
it’s 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)
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• Definition: Ordering
Whole Numbers:
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– Let a=n(A) and b=n(B)
then a<b (read a is less
than b) or b>a (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
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1. Counting chant – one, two, three, etc
2. Set Method – a set with three elements
can be matched with a proper subset of a
set with eight elements 3<8 and 8>3.
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Problem: determine which is greater
3 or 8 in three different ways (cont)
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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
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• Tally numeration system – single strokes,
one for each object counted.
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• Improved with grouping.
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The Egyptian Numeration System
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developed around 3400 B.C invovles grouping
by ten.
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=?
321.
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The Roman Numeration System
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• 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
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– Subtractive principle
– Multiplicative principle.
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Subtractive system
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• 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
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• Utilizes a horizontal bar above a numeral to
represent 1000 times the number
• Then V means 5 times 1000 or 5000
• and XI is 1100
• System still needs many more symbols than
current system and is cumbersome for doing
arithmetic.
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The Babylonian Numeration System
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• Evolved between 3000 and 2000 B.C.
• Used only two numerals, one and ten
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• 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.
Sections 2.3
The Hindu-Arabic System
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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 it’s own
value.
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Models for multi digit numbers
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• Bundles of Sticks – each ten sticks bound
together with a band
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• Base ten pieces (Dienes blocks) individual
cubes grouped in tens.
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The Hindu-Arabic System
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4. Additive and multiplicative
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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 one
Digits
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Numeral value 6x1000 + 5x100 + 2x10 + 3x1
Numeral
6523.
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Observations about the naming
procedure
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The number 0,1,…12 all have unique names
The numbers 13,14, …19 are the “teens”
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
The number 100, … 999 are combinations of hundreds
and previous names e.g. 637 reads “six hundred thirtyseven”
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
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• Three distinct ideas that children need to
learn to understand the Hindu-Arabic
numeration system .
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Base 5 operations
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• We can express numeration systems as base
systems
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– 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 system
e.g. base five 37five .
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