Readings for Session 7 – (Continued)
Order of Whole Numbers
Order:
The order
of whole numbers is a comparison of the relative size of sets
(comparing cardinal numbers).
For sets A and
B with cardinal
numbers a =
n(A)
and b = n(B),
if A is equivalent to a proper subset of
B, then we say that a is
less than b or
b is greater than
a.
Notation:
a <
b or b > a, these are read as
a
is less than b and
b is greater than a.
Example:
Pat sold several items on
eBay: $25, $36, $12, $21, $34, $22, $15, and $19.
Order the values in this list from least to greatest.
Solution:
$12,
$15, $19,
$21, $22,
$25, $34, $36
We illustrate the above solution
below where we use sets of dollar signs.
|
12 |
$$$$$$$$$$$$ |
Note that each set of dollars can be set up with a 1-1
correspondence with a
proper subset of the set
below it. |
|
15 |
$$$$$$$$$$$$$$$ |
|
|
19 |
$$$$$$$$$$$$$$$$$$$ |
|
|
21 |
$$$$$$$$$$$$$$$$$$$$$ |
|
|
22 |
$$$$$$$$$$$$$$$$$$$$$$ |
|
|
25 |
$$$$$$$$$$$$$$$$$$$$$$$$$ |
|
|
34 |
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |
|
|
36 |
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |
12 < 15 < 19 < 21 < 22 <
25 < 34 < 36
We need to pay attention to whether
we are asked to order the list from least to greatest or
greatest to least.
What do we mean by
≤ (less than or
equal to) and
≥
(greater than or equal to)? We modify the
definition for order for these cases. In this more general form
for ordering whole numbers, we use subset in the definition in
place of proper subset since subsets allow sets to be
equivalent.
Order (A more general definition):
For sets A
and B with cardinal
numbers a =
n(A)
and b = n(B),
if A is equivalent to a subset of
B, then we say that a
is less than or equal to b
or b is greater than or equal to
a.
Notation:
a
≤
b or
b
≥
a, these are read as
a
is less than or equal to b
and
b is greater than or equal to a.
Example:
Pat sold several items on
eBay: $22, $34, $12, $22, $34, $22, $12, and $19.
Order the values in this list from least to greatest.
Solution:
$12,
$12, $19,
$22, $22,
$22, $34, $34
We illustrate the above solution
below where we use sets of dollar signs.
|
12 |
$$$$$$$$$$$$ |
Note that each set of dollars can be set up with a 1-1
correspondence with a
subset of the set below it.
|
|
12 |
$$$$$$$$$$$$ |
|
|
19 |
$$$$$$$$$$$$$$$$$$$ |
|
|
22 |
$$$$$$$$$$$$$$$$$$$$$$ |
|
|
22 |
$$$$$$$$$$$$$$$$$$$$$$ |
|
|
22 |
$$$$$$$$$$$$$$$$$$$$$$ |
|
|
34 |
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |
|
|
34 |
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ |
Written with the
symbols: 12
≤ 12
≤ 19
≤ 22
≤ 22
≤ 22
≤ 34
≤ 34.
Also, note that
12
=
12
<
19
<
22
=
22
=
22
<
34
=
34
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