12-240/Classnotes for Thursday September 13: Difference between revisions
From Drorbn
Jump to navigationJump to search
No edit summary |
|||
| Line 24: | Line 24: | ||
'''F4:''' existence of inverses |
'''F4:''' existence of inverses |
||
<math>\forall \!\,</math> a <math>\in \!\,</math> F ,<math> \exists \!\ |
<math>\forall \!\,</math> a <math>\in \!\,</math> F ,<math> \exists \!\</math> c, d <math>\in \!\ </math> F such that a+c=o and a.d=1 |
||
| Line 30: | Line 30: | ||
<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F, a.(b+c)=a.b + a.c |
<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F, a.(b+c)=a.b + a.c |
||
== Examples == |
== Examples == |
||
Revision as of 11:20, 14 September 2012
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
In the second day of the class, the professor continues on the definition of a field.
Definition of a field
Combined with a part from the first class, we have a complete definition as follow:
A field is a set "F' with two binary operations +,x defind on it, and two special elements 0 ≠ 1 such that
F1: commutative law
a, b F: a+b=b+a and a.b=b.a
F2: associative law
a, b, c F: (a+b)+c=a+(b+c) and (a.b).c= a.(b.c)
F3: the existence of identity elements
a F, a+o=a and a.1=a
F4: existence of inverses
a F ,Failed to parse (syntax error): {\displaystyle \exists \!\} c, d F such that a+c=o and a.d=1
F5: contributive law
a, b, c F, a.(b+c)=a.b + a.c
Examples
Lecture Notes, upload by Starash
Page 1
Page 2
