Readings for Session 8 – (Continued)
Subtraction of Whole Numbers
How should we
define the subtraction of whole numbers?
In the lead-in example on an earlier page of this section, the remaining balance was the difference between the cardinalities of the sets for the checking account and the check. This also works for the third example (on the previous page) where
n(G) – n(H) = 3 – 3 = 0 = n(∅).
But, with the second example (on the previous page) the difference between the cardinalities does not give the expected result, e.g.,
n(A) – n(B) = 4 – 3 = 1 ≠ 0 = n(A – B).
In
this case, B is not a
subset of A. This
leads to the following set definition for subtraction of whole
numbers.
Set Definition for Subtraction of Whole Numbers: Let a = n(A) and b = n(B) where sets A and B are two finite sets such that B ⊆ A. Then a – b = n(A – B).
The result a – b is called the difference.
The
a is called the minuend
and the b is called
the subtrahend.
Investigation.
Does
the above definition apply to each of the following problems? If
not, how should we modify the definition for subtraction?
1.
A parent had $47 and gave $25 to a child. How
much cash did the parent have left?
2.
Lynn had to buy twelve books for this semester
and Pat had to buy nine books for this semester. How many more
books did Lynn have to buy than Pat?
3.
Sam has 483 songs on a MP3 player. How many
more songs does Sam need to have 1000 songs?
The first problem fits the
definition since the $25 given to the child is a subset of the
$47 the parent originally had. The parent has $22 left since 47
– 25 = 22.
The second problem does not fit the
above definition since Pat’s books are not a subset of Lynn’s
books. We would still consider the problem to be 12 – 9 = 3,
i.e., Lynn has three more books than Pat. But, Pat’s books can
be put into a one-to-one correspondence with a subset of Lynn’s
books, so with this modification the problem can be made to fit
a modified version of the above definition.
The third problem fits the
definition where songs Sam currently has on the MP3 player is a
subset of the songs Sam desires to have. Sam would like 517 more
songs since 1000 – 483 = 517. Note that this problem could be
thought of as an addition problem, 483 +
N = 1000, where
N represents the number of additional songs Sam would like to have.
This motivates the definition of subtraction of whole numbers to
be the inverse of addition that we mentioned at the beginning of
this section.
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