12-240/Classnotes for Tuesday September 11 - Revision history

Oguzhancan: /* Examples of Fields */

2012-09-21T19:16:58Z

Examples of Fields

← Older revision		Revision as of 15:16, 21 September 2012
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	*<math>\mathbb{F}_4</math> is '''not a field.'''		*<math>\mathbb{F}_4</math> is '''not a field.'''
	**It does not have the property <math>\mathbb{R}5</math>.		**It does not have the property <math>\mathbb{R}4</math>.
	:::<math>2 \cdot 0 = 0</math>		:::<math>2 \cdot 0 = 0</math>
	:::<math>2 \cdot 1 = 2</math>		:::<math>2 \cdot 1 = 2</math>

Oguzhancan: /* Existence of Negatives/Inverses */

2012-09-21T16:39:14Z

Existence of Negatives/Inverses

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	<math>\mathbb{R}4</math>		<math>\mathbb{R}4</math>
	:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a + b = 0\!</math>		:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a + b = 0\!</math>
	:<math>\forall\ a\ \epsilon\ \mathbb{R}\, a\ \neq 0, \exists\ b\ \epsilon\ \mathbb{R} \quad a \cdot b = 1\!</math>		:<math>\forall\ a\ \epsilon\ \mathbb{R}\ , a\ \neq 0 , \exists\ b\ \epsilon\ \mathbb{R} \quad a \cdot b = 1\!</math>

	=== Distributive Law ===		=== Distributive Law ===

Oguzhancan: /* Existence of Negatives/Inverses */

2012-09-21T16:38:43Z

Existence of Negatives/Inverses

← Older revision		Revision as of 12:38, 21 September 2012
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	<math>\mathbb{R}4</math>		<math>\mathbb{R}4</math>
	:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a + b = 0\!</math>		:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a + b = 0\!</math>
	:<math>\forall\ a\ \epsilon\ \mathbb{R}\ \exists\ b\ \epsilon\ \mathbb{R} \quad a \cdot b = 1\!</math>		:<math>\forall\ a\ \epsilon\ \mathbb{R}\, a\ \neq 0, \exists\ b\ \epsilon\ \mathbb{R} \quad a \cdot b = 1\!</math>

	=== Distributive Law ===		=== Distributive Law ===

Starash at 12:34, 14 September 2012

2012-09-14T12:34:50Z

← Older revision		Revision as of 08:34, 14 September 2012
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	[[Image:12-240-Sept11-Page8.jpeg\|250px]]		[[Image:12-240-Sept11-Page8.jpeg\|250px]]
	[[Image:12-240-Sept11-Page9.jpeg\|250px]]		[[Image:12-240-Sept11-Page9.jpeg\|250px]]

			== Lecture notes upload by [[User:Starash\|Starash]] ==
			Another upload of lecture 1 notes.
			<gallery>
			Image:Mat240-120911-p01.jpg\|page 1
			Image:Mat240-120911-p02.jpg\|page 2
			Image:Mat240-120911-p03.jpg\|page 3
			</gallery>

Drorbn at 20:51, 13 September 2012

2012-09-13T20:51:52Z

← Older revision		Revision as of 16:51, 13 September 2012
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	*Question: Are addition and multiplication defined here only arbitrary? Can we define many other ways to add or multiply, for a set, as long as the result satisfies F1-5 to show that F is indeed a field?		*Question: Are addition and multiplication defined here only arbitrary? Can we define many other ways to add or multiply, for a set, as long as the result satisfies F1-5 to show that F is indeed a field?
			** Answer by [[User:Drorbn\|Drorbn]] 16:51, 13 September 2012 (EDT): A "field" is a set with two operations and 0 and 1 so that some properties hold. In principle, the same set can be made into a field in many different ways - by choosing different operations (so long as they satisfy F1-5). In practice though, there is essentially only one field with 5 elements (but I the word "essentially" here requires an explanation). Many other sets can be considered as fields in many "genuinely different" ways, depending in how you choose the operations <math>+</math> and <math>\times</math>.

	*<math>\mathbb{F} = \mathbb{F}_3 = \mathbb{Z}/3 = \lbrace 0, 1, 2 \rbrace</math>		*<math>\mathbb{F} = \mathbb{F}_3 = \mathbb{Z}/3 = \lbrace 0, 1, 2 \rbrace</math>

Sina.zoghi: /* Examples of Fields */

2012-09-13T17:34:28Z

Examples of Fields

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	\|}		\|}

			*Question: Are addition and multiplication defined here only arbitrary? Can we define many other ways to add or multiply, for a set, as long as the result satisfies F1-5 to show that F is indeed a field?

	*<math>\mathbb{F} = \mathbb{F}_3 = \mathbb{Z}/3 = \lbrace 0, 1, 2 \rbrace</math>		*<math>\mathbb{F} = \mathbb{F}_3 = \mathbb{Z}/3 = \lbrace 0, 1, 2 \rbrace</math>

Drorbn: /* Scanned Notes by Sina.zoghi */

2012-09-13T14:50:45Z

Scanned Notes by Sina.zoghi

← Older revision		Revision as of 10:50, 13 September 2012
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	==Scanned Notes by [[User:Sina.zoghi\|Sina.zoghi]]==		==Scanned Notes by [[User:Sina.zoghi\|Sina.zoghi]]==
	[[User:Sina.zoghi\|Sina.zoghi]] - I ~~have~~ ~~linked~~ ~~your notes below, so that you and other can see how such scanned notes may be linked. Though I'm afraid~~ the ~~quality of the~~ scans ~~is rather low~~ - ~~they are of low contrast and~~ there is too much "white space" around each page. So ~~the~~ ~~thumbnails~~ ~~are~~ ~~barely~~ ~~readable.~~ If ~~you'd~~ be ~~able~~ to ~~re-upload better~~ scans ~~(under~~ ~~the~~ ~~same~~ ~~filenames)~~ ~~that~~ ~~will~~ be ~~great~~. [[User:Drorbn\|Drorbn]] 09:14, 12 September 2012 (EDT)		[[User:Sina.zoghi\|Sina.zoghi]] - Thanks for improving on the previously-uploaded scans - though there is still too much "white space" around each page. It is probably not worth your while to fix it for these scans, but it is something to keep in mind for later ones. [[User:Drorbn\|Drorbn]] 10:50, 13 September 2012 (EDT)

	[[Image:12-240-Sept11-Page1.jpeg\|250px]]		[[Image:12-240-Sept11-Page1.jpeg\|250px]]

Drorbn: /* Scanned Notes by User:Sina.zoghi */

2012-09-12T15:27:18Z

Scanned Notes by User:Sina.zoghi

← Older revision		Revision as of 11:27, 12 September 2012
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	:<math>\Box</math>		:<math>\Box</math>

	==Scanned Notes by [[User:Sina.zoghi]]==		==Scanned Notes by [[User:Sina.zoghi\|Sina.zoghi]]==
	[[User:Sina.zoghi]] - I have linked your notes below, so that you and other can see how such scanned notes may be linked. Though I'm afraid the quality of the scans is rather low - they are of low contrast and there is too much "white space" around each page. So the thumbnails are barely readable. If you'd be able to re-upload better scans (under the same filenames) that will be great. [[User:Drorbn\|Drorbn]] 09:14, 12 September 2012 (EDT)		[[User:Sina.zoghi\|Sina.zoghi]] - I have linked your notes below, so that you and other can see how such scanned notes may be linked. Though I'm afraid the quality of the scans is rather low - they are of low contrast and there is too much "white space" around each page. So the thumbnails are barely readable. If you'd be able to re-upload better scans (under the same filenames) that will be great. [[User:Drorbn\|Drorbn]] 09:14, 12 September 2012 (EDT)

	[[Image:12-240-Sept11-Page1.jpeg\|250px]]		[[Image:12-240-Sept11-Page1.jpeg\|250px]]

Drorbn: /* Scanned Notes by User:Sina.zoghi */

2012-09-12T13:14:31Z

Scanned Notes by User:Sina.zoghi

← Older revision		Revision as of 09:14, 12 September 2012
Line 267:		Line 267:

	==Scanned Notes by [[User:Sina.zoghi]]==		==Scanned Notes by [[User:Sina.zoghi]]==
	[[User:Sina.zoghi]] - I have linked your notes below, so that you and other can see how such scanned notes may be linked. Though I'm afraid the quality of the scans is rather low - they are of low contrast and there is too much "white space" around each page. So the thumbnails are barely readable. If you'd be able to re-upload better scans (under the same filenames) that will be great.		[[User:Sina.zoghi]] - I have linked your notes below, so that you and other can see how such scanned notes may be linked. Though I'm afraid the quality of the scans is rather low - they are of low contrast and there is too much "white space" around each page. So the thumbnails are barely readable. If you'd be able to re-upload better scans (under the same filenames) that will be great. [[User:Drorbn\|Drorbn]] 09:14, 12 September 2012 (EDT)

	[[Image:12-240-Sept11-Page1.jpeg\|250px]]		[[Image:12-240-Sept11-Page1.jpeg\|250px]]

Drorbn: /* Scanned Notes by User:Sina.zoghi */

2012-09-12T13:14:16Z

Scanned Notes by User:Sina.zoghi

← Older revision		Revision as of 09:14, 12 September 2012
Line 267:		Line 267:

	==Scanned Notes by [[User:Sina.zoghi]]==		==Scanned Notes by [[User:Sina.zoghi]]==
			[[User:Sina.zoghi]] - I have linked your notes below, so that you and other can see how such scanned notes may be linked. Though I'm afraid the quality of the scans is rather low - they are of low contrast and there is too much "white space" around each page. So the thumbnails are barely readable. If you'd be able to re-upload better scans (under the same filenames) that will be great.
	<gallery>

⚫	Image:12-240-Sept11-~~Page1~~.jpeg
	Image:12-240-Sept11-~~Page2~~.jpeg		[[Image:12-240-Sept11-Page1.jpeg\|250px]]
	Image:12-240-Sept11-~~Page3~~.jpeg		[[Image:12-240-Sept11-Page2.jpeg\|250px]]
	Image:12-240-Sept11-~~Page4~~.jpeg		[[Image:12-240-Sept11-Page3.jpeg\|250px]]
	Image:12-240-Sept11-~~Page5~~.jpeg		[[Image:12-240-Sept11-Page4.jpeg\|250px]]
	Image:12-240-Sept11-~~Page6~~.jpeg		[[Image:12-240-Sept11-Page5.jpeg\|250px]]
	Image:12-240-Sept11-~~Page7~~.jpeg		[[Image:12-240-Sept11-Page6.jpeg\|250px]]
	Image:12-240-Sept11-~~Page8~~.jpeg		[[Image:12-240-Sept11-Page7.jpeg\|250px]]
	Image:12-240-Sept11-~~Page9~~.jpeg		[[Image:12-240-Sept11-Page8.jpeg\|250px]]
		⚫	[[Image:12-240-Sept11-Page9.jpeg\|250px]]
	</gallery>

← Older revision		Revision as of 08:34, 14 September 2012
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	[[Image:12-240-Sept11-Page8.jpeg\|250px]]		[[Image:12-240-Sept11-Page8.jpeg\|250px]]
	[[Image:12-240-Sept11-Page9.jpeg\|250px]]		[[Image:12-240-Sept11-Page9.jpeg\|250px]]

			== Lecture notes upload by [[User:Starash\|Starash]] ==
			Another upload of lecture 1 notes.
			<gallery>
			Image:Mat240-120911-p01.jpg\|page 1
			Image:Mat240-120911-p02.jpg\|page 2
			Image:Mat240-120911-p03.jpg\|page 3
			</gallery>

← Older revision		Revision as of 10:50, 13 September 2012
Line 267:		Line 267:

	==Scanned Notes by [[User:Sina.zoghi\|Sina.zoghi]]==		==Scanned Notes by [[User:Sina.zoghi\|Sina.zoghi]]==
	[[User:Sina.zoghi\|Sina.zoghi]] - I ~~have~~ ~~linked~~ ~~your notes below, so that you and other can see how such scanned notes may be linked. Though I'm afraid~~ the ~~quality of the~~ scans ~~is rather low~~ - ~~they are of low contrast and~~ there is too much "white space" around each page. So ~~the~~ ~~thumbnails~~ ~~are~~ ~~barely~~ ~~readable.~~ If ~~you'd~~ be ~~able~~ to ~~re-upload better~~ scans ~~(under~~ ~~the~~ ~~same~~ ~~filenames)~~ ~~that~~ ~~will~~ be ~~great~~. [[User:Drorbn\|Drorbn]] 09:14, 12 September 2012 (EDT)		[[User:Sina.zoghi\|Sina.zoghi]] - Thanks for improving on the previously-uploaded scans - though there is still too much "white space" around each page. It is probably not worth your while to fix it for these scans, but it is something to keep in mind for later ones. [[User:Drorbn\|Drorbn]] 10:50, 13 September 2012 (EDT)

	[[Image:12-240-Sept11-Page1.jpeg\|250px]]		[[Image:12-240-Sept11-Page1.jpeg\|250px]]