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

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* PDF notes by [[User:Harbansb]]: [[Media:06-240-0912.pdf|September 12 Notes]].
* 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.
* 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
* [http://www.yousendit.com/transfer.php?action=download
* 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:
** [[06-240: Edit1.jpg]]
** [[06-240: Edit2.jpg]]

=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:<br>
<math>\mathbb{R}1:\forall a, b\in \mathbb{R}\mbox{ s.th.} \quad a+b=b+a \quad \mbox{and} \quad a\cdot b=b\cdot a</math> (The Commutative Laws)<br>
<math>\mathbb{R}2:\forall a, b, c\in \mathbb{R}\mbox{ s.th.} \quad (a+b)+c=a+(b+c) \quad \mbox{and} \quad (a\cdot b)\cdot c=a\cdot (b\cdot c) </math> (The Associative Laws)<br>
<math>\mathbb{R}3:0\mbox{ is an additive unit} \quad \mbox{and} \quad 1\mbox{ is a multiplicative unit}</math> (The Existence of Units/Identities)<br>
<math>\mathbb{R}4:\forall a\in \mathbb{R} \ \exists b\in \mathbb(R) \mbox{ s.th.} \ a+b=0</math>

This is incomplete.

Revision as of 12:19, 13 June 2007

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