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

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* PDF notes by [[User:Alla]]: [[Media:MAT_Lect001.pdf|Week 1 Lecture 1 notes]]
 
* 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:
 
* 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]]
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::[[Image:Edit1.jpg|200px]] [[Image:Edit2.jpg|200px]]
**  [[06-240: Edit2.jpg]]
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=Notes=
 
=Notes=
  
 
==The Real Numbers==
 
==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>
+
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:\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 takes way too long. It is probably more practical to type the notes in Word and upload it onto the site, if we want a typesetted version of the notes.
+
<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.