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2.2 Some Examples
2.35
Example (Calculator operations.)
Let
![$\widetilde C$](img332.gif)
denote the set of all numbers that can be entered into your
calculator. The exact composition of
![$\widetilde C$](img332.gif)
depends on the model of your
calculator. Let
![$C=\widetilde C\cup\{E\}$](img333.gif)
where
![$E$](img334.gif)
is some object not in
![$\widetilde C$](img332.gif)
. I
will call
the error. I think of
![$E$](img334.gif)
as the result produced when you enter
![$1/0$](img335.gif)
. Define four binary operations
![$\oplus,\ominus ,\odot$](img336.gif)
, and
![$\oslash$](img337.gif)
on
![$C$](img338.gif)
by
On my calculator
If
![$\circ$](img239.gif)
denotes any of
![$\oplus,\ominus,\odot,\oslash$](img341.gif)
, I define
On all calculators with which I am familiar,
![$\oplus$](img343.gif)
and
![$\odot$](img344.gif)
are commutative operations,
![$0$](img246.gif)
is an
identity for
![$\oplus$](img343.gif)
,
![$1$](img17.gif)
is an identity for
![$\odot$](img344.gif)
, and every element of
![$C$](img338.gif)
except for
![$E$](img334.gif)
is invertible for
![$\oplus$](img343.gif)
. On my calculator
![$\displaystyle 1\oslash 3$](img345.gif) |
![$\textstyle =$](img346.gif) |
![$\displaystyle 0.333333333$](img347.gif) |
(2.36) |
![$\displaystyle 0.333333333 \odot 3$](img348.gif) |
![$\textstyle =$](img346.gif) |
![$\displaystyle 0.999999999$](img349.gif) |
(2.37) |
![$\displaystyle 0.333333333 \odot 3.000000003$](img350.gif) |
![$\textstyle =$](img346.gif) |
![$\displaystyle 1$](img351.gif) |
(2.38) |
![$\displaystyle 0.333333333 \odot 3.000000004$](img352.gif) |
![$\textstyle =$](img346.gif) |
![$\displaystyle 1.$](img353.gif) |
(2.39) |
Thus
![$0.333333333$](img354.gif)
has two different inverses! It follows from theorem
2.15 that
![$\odot$](img344.gif)
is not associative. Your calculator may give different
results for the calculations (
2.38) and (
2.39) but none of the
calculator operations are associative.
2.40
Exercise.
A
Verify that calculator addition
![$(\oplus)$](img355.gif)
and calculator
multiplication
![$(\odot)$](img356.gif)
are not associative, by finding calculator numbers
![$a$](img24.gif)
,
![$b$](img42.gif)
,
![$c$](img199.gif)
,
![$x$](img85.gif)
,
![$y$](img86.gif)
, and
![$z$](img289.gif)
such that
![$a \oplus(b\oplus c) \neq (a\oplus b)\oplus c$](img357.gif)
, and
![$x\odot(y\odot z) \neq (x\odot y)\odot z$](img358.gif)
.
2.41
Notation.
If
![$n\in\mbox{{\bf N}}$](img359.gif)
, let
Hence, for example
2.42
Definition (
.)
Let
![$n\in\mbox{{\bf N}}$](img359.gif)
, with
![$n\geq 2$](img364.gif)
. We define two binary operations
![$\oplus_n$](img365.gif)
and
![$\odot_n$](img366.gif)
on
![$\mbox{{\bf Z}}_n$](img363.gif)
by:
for all
,
and for all
,
Thus,
and
The operations
and
are both commutative (since
and
are commutative on
). Clearly
is an identity for
, and
is an
identity for
. Every element of
is invertible for
and
2.43
Definition (Multiplication table.)
Let
![$\circ$](img239.gif)
be a binary operation on a finite set
![$A=\{a_1,a_2,\cdots,a_n\}$](img373.gif)
having
![$n$](img320.gif)
elements. We construct a
multiplication table for
![$\circ$](img239.gif)
as follows: We
write down a table with
![$n$](img320.gif)
rows and
![$n$](img320.gif)
columns. Along the top of the table we list
the elements of
![$A$](img50.gif)
as labels for the columns. Along the left side of the table we
list the elements of
![$A$](img50.gif)
(in the same order) as labels for the rows. (See the figure
to see what is meant by this.) If
![$(x,y)\in A^2$](img374.gif)
, we write the product
![$x\circ y$](img375.gif)
in
the box of our table whose row label is
![$x$](img85.gif)
and whose column label is
![$y$](img86.gif)
.
Multiplication table for
![$\circ$](img239.gif)
2.44
Examples.
Below are the multiplication tables for
![$\oplus_5$](img389.gif)
and
![$\odot_5$](img390.gif)
:
![$\oplus_5$](img389.gif) |
0 |
1 |
2 |
3 |
4 |
0 |
0 |
1 |
2 |
3 |
4 |
1 |
1 |
2 |
3 |
4 |
0 |
2 |
2 |
3 |
4 |
0 |
1 |
3 |
3 |
4 |
0 |
1 |
2 |
4 |
4 |
0 |
1 |
2 |
3 |
![$\odot_5$](img390.gif) |
0 |
1 |
2 |
3 |
4 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
2 |
3 |
4 |
2 |
0 |
2 |
4 |
1 |
3 |
3 |
0 |
3 |
1 |
4 |
2 |
4 |
0 |
4 |
3 |
2 |
1 |
By looking at the multiplication table for
we see that
Hence all the non-zero elements of
have inverses under
.
Both of the operations
and
are associative. This follows from
the fact that
and
are associative operations on
, by a
straightforward but lengthy argument. The details are given in appendix
B.
2.45
Exercise.
Write down the multiplication table for
![$\odot_6$](img393.gif)
on
![$\mbox{{\bf Z}}_6$](img394.gif)
. Determine which
elements of
![$\mbox{{\bf Z}}_6$](img394.gif)
are invertible for
![$\odot_6$](img393.gif)
, and find the inverse for each
invertible element.
2.46
Exercise.
Let
![$\{x,y,z\}$](img395.gif)
be a set containing three distinct elements.
![$(x\neq y$](img396.gif)
,
![$y\neq z$](img397.gif)
,
![$z\neq x)$](img398.gif)
.
Let
![$\circ$](img239.gif)
be the binary operation on
![$\{x,y,z\}$](img395.gif)
determined by the
multiplication table:
- a)
- Show that there is an identity element for
. (Which of
is the
identity?)
- b)
- Show that
has two different inverses for
.
- c)
- Explain why the result of part b does not contradict the theorem on uniqueness
of inverses.
2.47
Note.
An early example of a binary operation that was not an obvious
generalization of one of the operations
![$+,-,\cdot,/$](img399.gif)
on numbers was the use of union
and intersection as binary operations on the set of all sets by George
Boole[
11]. In
Laws of Thought (1854),
Boole introduces the operation
![$+$](img274.gif)
(for union) and
![$\times$](img202.gif)
(for intersection) on `` classes" (although he
usually writes
![$xy$](img400.gif)
instead of
![$x \times y$](img401.gif)
). He explicitly states
which he calls commutative and distributive laws. He does not mention associativity,
and writes
![$xyz$](img403.gif)
without parentheses. He denotes `` Nothing" by
![$0$](img246.gif)
and ``
the Universe" by
![$1$](img17.gif)
, and notes that
![$0$](img246.gif)
and
![$1$](img17.gif)
have the usual properties. As an
example of the distributive law, Boole gives
European men and women
![$=$](img143.gif)
European men
and European women.
Boole's
is not really a binary operation since he only
defines
when
and
have no elements in common.
The word associative, in its mathematical sense, was introduced by
William Hamilton[24, p114] in
1843
in a paper on quaternions. According to
[14, p284], the words commutative and distributive were
introduced by Francois -Joseph Servois in 1813.
Next: 2.3 The Field Axioms
Up: 2. Fields
Previous: 2.1 Binary Operations
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