06-240/Classnotes For Tuesday, September 12 - Revision history

Drorbn at 22:13, 11 July 2007

2007-07-11T22:13:45Z

← Older revision		Revision as of 18:13, 11 July 2007
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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]]		::[[Image:Edit1.jpg\|200px]] [[Image:Edit2.jpg\|200px]]
	** [[06-240: Edit2.jpg]]

	=Notes=		=Notes=

Drorbn: /* The Real Numbers */

2007-07-11T22:09:16Z

The Real Numbers

← Older revision		Revision as of 18:09, 11 July 2007
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	==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 · (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 · (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}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}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>		<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.		This is incomplete.

Drorbn: Reverted edit of 218.225.182.47, changed back to last version by 70.24.245.27

2007-06-13T16:33:15Z

Reverted edit of 218.225.182.47, changed back to last version by 70.24.245.27

@@ Line 3: / Line 3: @@
 * 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
+* [http://www.yousendit.com/transfer.php?action=download&ufid=38FF36BF7ED1E1BA September 12 Notes] for re-uploading, please email at jeff.matskin@utoronto.ca

218.225.182.47 at 16:19, 13 June 2007

2007-06-13T16:19:38Z

@@ Line 3: / Line 3: @@
 * 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
+* [http://www.yousendit.com/transfer.php?action=download

70.24.245.27 at 19:45, 8 October 2006

2006-10-08T19:45:22Z

← Older revision		Revision as of 15:45, 8 October 2006
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	<math>\mathbb{R}4:\forall a\in \mathbb{R} \ \exists b\in \mathbb(R) \mbox{ s.th.} \ a+b=0</math>		<math>\mathbb{R}4:\forall a\in \mathbb{R} \ \exists b\in \mathbb(R) \mbox{ s.th.} \ a+b=0</math>

			This is incomplete.
	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.

Alla: changed file name

2006-09-25T02:30:29Z

changed file name

← Older revision		Revision as of 22:30, 24 September 2006
Line 4:		Line 4:
	* 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&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]].		* 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]]		** [[06-240: Edit1.jpg]]

Alla: Posted Week 1 Lecture 1 notes

2006-09-25T01:59:04Z

Posted Week 1 Lecture 1 notes

← Older revision		Revision as of 21:59, 24 September 2006
Line 4:		Line 4:
	* 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&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]].
	* 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]]		** [[06-240: Edit1.jpg]]

Wanmike at 03:04, 23 September 2006

2006-09-23T03:04:16Z

← Older revision		Revision as of 23:04, 22 September 2006
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	==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 · (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 · (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} \quad a+b=b+a \quad \mbox{and} \quad a\cdot b=b\cdot a</math> (The Commutative Laws)<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} \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)		<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.

Wanmike at 02:53, 23 September 2006

2006-09-23T02:53:10Z

← Older revision		Revision as of 22:53, 22 September 2006
Line 11:		Line 11:

	==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 · (times) and two special elements: 0 (zero) and 1 (one), such that the following laws hold true: <p>		The Real Numbers are a set (denoted by <math>\mathbb{R}</math>) along with two binary operations: + (plus) and · (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} \quad a~~\cdot~~ b=b~~\cdot~~ a \quad \mbox{and} \quad a+b=b+a</math> (The Commutative Laws)		<math>\mathbb{R}1:\forall a, b\in \mathbb{R} \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} \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)

70.24.245.27 at 02:46, 23 September 2006

2006-09-23T02:46:24Z

← Older revision		Revision as of 22:46, 22 September 2006
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	** [[06-240: Edit1.jpg]]		** [[06-240: Edit1.jpg]]
	** [[06-240: Edit2.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 · (times) and two special elements: 0 (zero) and 1 (one), such that the following laws hold true: <p>
			<math>\mathbb{R}1 :\forall a, b\in \mathbb{R} \quad a\cdot b=b\cdot a \quad \mbox{and} \quad a+b=b+a</math> (The Commutative Laws)