12-240/Classnotes for Thursday September 13

#	Week of...	Notes and Links
Additions to this web site no longer count towards good deed points.
1	Sep 10	About This Class, Tuesday, Thursday
2	Sep 17	HW1, Tuesday, Thursday, HW1 Solutions
3	Sep 24	HW2, Tuesday, Class Photo, Thursday
4	Oct 1	HW3, Tuesday, Thursday
5	Oct 8	HW4, Tuesday, Thursday
6	Oct 15	Tuesday, Thursday
7	Oct 22	HW5, Tuesday, Term Test was on Thursday. HW5 Solutions
8	Oct 29	Why LinAlg?, HW6, Tuesday, Thursday, Nov 4 is the last day to drop this class
9	Nov 5	Tuesday, Thursday
10	Nov 12	Monday-Tuesday is UofT November break, HW7, Thursday
11	Nov 19	HW8, Tuesday,Thursday
12	Nov 26	HW9, Tuesday , Thursday
13	Dec 3	Tuesday UofT Fall Semester ends Wednesday
F	Dec 10	The Final Exam (time, place, style, office hours times)
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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

$\forall \!\,$ a, b $\in \!\,$ F: a+b=b+a and a.b=b.a

F2: associative law

$\forall \!\,$ a, b, c $\in \!\,$ F: (a+b)+c=a+(b+c) and (a.b).c= a.(b.c)

F3: the existence of identity elements

$\forall \!\,$ a $\in \!\,$ F, a+o=a and a.1=a

F4: existence of inverses

$\forall \!\,$ a $\in \!\,$ F , $\exists \!\,$ c, d $\in \!\$ F such that a+c=o and a.d=1

F5: contributive law

$\forall \!\,$ a, b, c $\in \!\,$ F, a.(b+c)=a.b + a.c

Identity uniqueness It makes sense to define an operation -: F -> F called "negation"

For a $\in \!\,$ F define -a to be equal that b $\in \!\,$ F for which a+b=0, i.e, a+(-a)=0

Ex: F(5)={0,1,2,3,4}, define +,x

Question: What is(-3)?

Answer: -3=2