Field Axioms
Definition 1.
A
binary operation
,
*
, on a set
A
, is a function
*
:
A
×
A
→
A
Note:
• *
has to be defined for every pair
a, b
∈
A
, and give an element
of
A
•
input is an ordered pair
•
the value on an ordered pair (
a, b
) is written
a
*
b
Example.
On
Z
, +,

, and
·
, but not
÷
.
Definition 2.
If
*
is a binary operation on
A
, then
• *
is
commutative
if
a
*
b
=
b
*
a
for all
a, b
∈
A
• *
is
associative
if
a
*
(
b
*
c
) = (
a
*
b
)
*
c
for all
a, b, c
∈
A
• *
has an
identity element
e
if
a
*
e
=
e
*
a
=
a
for all
a
∈
A
•
if
*
has an identity
e
and
a
∈
A
, we say that
a
has an inverse
if
there exists
b
∈
A
s.t.
a
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 Fall '09
 Linear Algebra, Algebra, binary operations, Identity element, Binary operation, ﬁeld axioms

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