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2.2 Some Examples
2.35
Example (Calculator operations.)
Let

denote the set of all numbers that can be entered into your
calculator. The exact composition of

depends on the model of your
calculator. Let

where

is some object not in

. I
will call
the error. I think of

as the result produced when you enter

. Define four binary operations

, and

on

by
On my calculator
If

denotes any of

, I define
On all calculators with which I am familiar,

and

are commutative operations,

is an
identity for

,

is an identity for

, and every element of

except for

is invertible for

. On my calculator
 |
 |
 |
(2.36) |
 |
 |
 |
(2.37) |
 |
 |
 |
(2.38) |
 |
 |
 |
(2.39) |
Thus

has two different inverses! It follows from theorem
2.15 that

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

and calculator
multiplication

are not associative, by finding calculator numbers

,

,

,

,

, and

such that

, and

.
2.41
Notation.
If

, let
Hence, for example
2.42
Definition (
.)
Let

, with

. We define two binary operations

and

on

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

be a binary operation on a finite set

having

elements. We construct a
multiplication table for

as follows: We
write down a table with

rows and

columns. Along the top of the table we list
the elements of

as labels for the columns. Along the left side of the table we
list the elements of

(in the same order) as labels for the rows. (See the figure
to see what is meant by this.) If

, we write the product

in
the box of our table whose row label is

and whose column label is

.
Multiplication table for

2.44
Examples.
Below are the multiplication tables for

and

:
 |
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 |
 |
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

on

. Determine which
elements of

are invertible for

, and find the inverse for each
invertible element.
2.46
Exercise.
Let

be a set containing three distinct elements.

,

,

.
Let

be the binary operation on

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

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

(for union) and

(for intersection) on `` classes" (although he
usually writes

instead of

). He explicitly states
which he calls commutative and distributive laws. He does not mention associativity,
and writes

without parentheses. He denotes `` Nothing" by

and ``
the Universe" by

, and notes that

and

have the usual properties. As an
example of the distributive law, Boole gives
European men and women

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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