14-240/Classnotes for Monday September 15 - Revision history

Boyang.wu: /* Scanned Lecture Notes by AM */

2014-12-08T05:57:02Z

Scanned Lecture Notes by AM

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			==Scanned Lecture Notes by [[User Boyang.wu\|Boyang.wu]]==

	[[File:W21.pdf]]		[[File:W21.pdf]]

Boyang.wu: /* Scanned Lecture Notes by AM */

2014-12-08T05:23:20Z

Scanned Lecture Notes by AM

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	File:W21.pdf		[[File:W21.pdf]]

Boyang.wu: /* Scanned Lecture Notes by AM */

2014-12-08T05:23:03Z

Scanned Lecture Notes by AM

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Hameeral: Format scanned lecture notes into a gallery.

2014-09-29T00:58:45Z

Format scanned lecture notes into a gallery.

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	==Scanned ~~notes~~==		==Scanned Lecture Notes by [[User:AM\|AM]]==

	http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2)
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	http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)
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AM: Undo revision 13367 by AM (talk)

2014-09-22T16:16:04Z

Undo revision 13367 by AM (talk)

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			==Definition of Subtraction and Division==
			* Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>.
			* Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>.

	==Basic Properties of a Field (cont'd)==		==Basic Properties of a Field (cont'd)==

AM: /* Definition of Subtraction and Division */

2014-09-22T16:13:58Z

Definition of Subtraction and Division

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	==Definition of Subtraction and Division==
	* Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>.
	* Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>.

	==Basic Properties of a Field (cont'd)==		==Basic Properties of a Field (cont'd)==

Penlong: /* Scanned notes */

2014-09-21T15:07:31Z

Scanned notes

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	</math>		</math>

	==Scanned notes===		==Scanned notes==
	http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2)		http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2)
	http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)		http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)

Penlong: /* Theorem */

2014-09-21T15:07:15Z

Theorem

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	\end{align}		\end{align}
	</math>		</math>

			==Scanned notes===
	http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2)		http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2)
	http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)		http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)

Penlong at 15:05, 21 September 2014

2014-09-21T15:05:56Z

Show changes

Yue.Jiang at 03:43, 18 September 2014

2014-09-18T03:43:33Z

← Older revision		Revision as of 23:43, 17 September 2014
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	* 8. <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.		* 8. <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.
	proof of 8: By F3 , <math>a \times 0 = a \times (0 + 0)</math>;		*proof of 8: By F3 , <math>a \times 0 = a \times (0 + 0)</math>
	By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;		By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;
	By F3 , <math>a \times 0 = 0 + a \times 0</math>;		By F3 , <math>a \times 0 = 0 + a \times 0</math>;
	By Thm P1,<math>0 = a \times 0</math>.		By Thm P1,<math>0 = a \times 0</math>.

	9. <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;		* 9. <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;
	<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.		<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.
	proof of 9: By F3 , <math>\times b = 0 \neq 1</math>.		proof of 9: By F3 , <math>\times b = 0 \neq 1</math>.

	10. <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.		* 10. <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.

	11. <math>(-a) \times (-b) = a \times b</math>.		* 11. <math>(-a) \times (-b) = a \times b</math>.

	12. <math>a \times b = 0 \iff a = 0 or b = 0</math>.		* 12. <math>a \times b = 0 \iff a = 0 or b = 0</math>.
	proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a \times b = 0 \times b = 0</math>;		proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a \times b = 0 \times b = 0</math>;
	By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.		By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.