Difference between revisions of "06-240/Classnotes For Tuesday, September 12"

From Drorbn
Jump to: navigation, search
 
 
(13 intermediate revisions by 7 users not shown)
Line 1: Line 1:
 
{{06-240/Navigation}}
 
{{06-240/Navigation}}
  
* PDF notes by [[User:Harbansb]]: [[Media:06-240-0912.pdf]].
+
* PDF notes by [[User:Harbansb]]: [[Media:06-240-0912.pdf|September 12 Notes]].
 +
* If I have made an error in my notes, or you would like the editable OpenOffice file, feel free to e-mail me at harbansb@msn.com.
 +
* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca
 +
* PDF notes by [[User:Alla]]: [[Media:MAT_Lect001.pdf|Week 1 Lecture 1 notes]]
 +
* Below are a couple of lemmata critical to the derivation we did in class - the Professor left this little work to the students:
 +
::[[Image:Edit1.jpg|200px]] [[Image:Edit2.jpg|200px]]
 +
 
 +
=Notes=
 +
 
 +
==The Real Numbers==
 +
The Real Numbers are a set (denoted by <math>\mathbb{R}</math>) along with two binary operations: + (plus) and &middot; (times) and two special elements: 0 (zero) and 1 (one), such that the following laws hold true:
 +
 
 +
<math>\mathbb{R}1</math>: <math>\forall a, b\in \mathbb{R}</math> we have <math>a+b=b+a</math> and <math>a\cdot b=b\cdot a</math> (The Commutative Laws)
 +
 
 +
<math>\mathbb{R}2</math>: <math>\forall a, b, c\in \mathbb{R}</math> we have <math>(a+b)+c=a+(b+c)</math> and <math>(a\cdot b)\cdot c=a\cdot (b\cdot c)</math> (The Associative Laws)
 +
 
 +
<math>\mathbb{R}3</math>: <math>0</math> is an additive unit and <math>1</math> is a multiplicative unit (The Existence of Units/Identities)
 +
 
 +
<math>\mathbb{R}4</math>: <math>\forall a\in \mathbb{R} \ \exists b\in \mathbb{R} \mbox{ s.t.} \ a+b=0</math>
 +
 
 +
This is incomplete.

Latest revision as of 18:13, 11 July 2007

  • PDF notes by User:Harbansb: September 12 Notes.
  • If I have made an error in my notes, or you would like the editable OpenOffice file, feel free to e-mail me at harbansb@msn.com.
  • September 12 Notes for re-uploading, please email at jeff.matskin@utoronto.ca
  • PDF notes by User:Alla: Week 1 Lecture 1 notes
  • Below are a couple of lemmata critical to the derivation we did in class - the Professor left this little work to the students:
Edit1.jpg Edit2.jpg

Notes

The Real Numbers

The Real Numbers are a set (denoted by \mathbb{R}) along with two binary operations: + (plus) and · (times) and two special elements: 0 (zero) and 1 (one), such that the following laws hold true:

\mathbb{R}1: \forall a, b\in \mathbb{R} we have a+b=b+a and a\cdot b=b\cdot a (The Commutative Laws)

\mathbb{R}2: \forall a, b, c\in \mathbb{R} we have (a+b)+c=a+(b+c) and (a\cdot b)\cdot c=a\cdot (b\cdot c) (The Associative Laws)

\mathbb{R}3: 0 is an additive unit and 1 is a multiplicative unit (The Existence of Units/Identities)

\mathbb{R}4: \forall a\in \mathbb{R} \ \exists b\in \mathbb{R} \mbox{ s.t.} \ a+b=0

This is incomplete.