12-240/Classnotes for Thursday September 13 - Revision history

Failure at 06:01, 23 October 2012

2012-10-23T06:01:02Z

2012-10-23T06:00:50Z

2012-09-14T17:23:09Z

Theorems

← Older revision		Revision as of 13:23, 14 September 2012
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	if a+c=b+c, then a=b		if a+c=b+c, then a=b

	if a.c=b.c and c<math>\ne \!\, ~~then~~ </math>0, a=b		if a.c=b.c and c<math>\ne \!\,</math>0, then a=b

	'''Theorem 2: Identity uniqueness'''		'''Theorem 2: Identity uniqueness'''

2012-09-14T17:22:20Z

Theorems

← Older revision		Revision as of 13:22, 14 September 2012
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	== Theorems ==		== Theorems ==

	'''Cancellation laws'''		'''Theorem 1: Cancellation laws'''
	<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F		<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F

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	if a.c=b.c and c<math>\ne \!\, then </math>0, a=b		if a.c=b.c and c<math>\ne \!\, then </math>0, a=b

	'''Identity uniqueness'''		'''Theorem 2: Identity uniqueness'''

	Identity elements 0 and 1 mentioned in '''F3''' are unique		Identity elements 0 and 1 mentioned in '''F3''' are unique
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	if a.b=a and a.b'=a and a<math>\ne\!\,</math>0, then b=b'=1		if a.b=a and a.b'=a and a<math>\ne\!\,</math>0, then b=b'=1

	'''Inverse uniqueness'''		'''Theorem 3: Inverse uniqueness'''

	Elements c and d mentioned in '''F4''' are unique		Elements c and d mentioned in '''F4''' are unique
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	if a.b=1 and a.b'=1, then b=b'		if a.b=1 and a.b'=1, then b=b'

			'''Theorem 4'''

			<math>\forall \!\,</math> a <math>\in \!\,</math> F

			-( -a) = a

			'''Theorem 4'''

			<math>\forall \!\,</math> a <math>\in \!\,</math> F

			0.a= 0

	== Significance ==		== Significance ==

2012-09-14T17:13:54Z

Significance

← Older revision		Revision as of 13:13, 14 September 2012
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	== Significance ==		== Significance ==

	'''~~Identity~~ uniqueness'''		'''inverse uniqueness'''

	It makes sense to define an operation		It makes sense to define an operation
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	Ex: F(5)={0,1,2,3,4}, define +,x		Ex: F(5)={0,1,2,3,4}, define +,x

	Question: What is(-3)?		Question 1: What is(-3)?

	Answer: -3=2		Answer: -3=2 and -3 is unique

			Similarly, the inverse uniqueness also makes sense a^(-1)


			''' identity uniqueness'''

	== Lecture Notes, upload by [[User:Starash\|Starash]] ==		== Lecture Notes, upload by [[User:Starash\|Starash]] ==

2012-09-14T17:09:52Z

Theorems

@@ Line 37: / Line 37: @@
 if a+c=b+c, then a=b
-if ac=bc and c<math>\ne \!\,</math>0, a=b
+if a.c=b.c and c<math>\ne \!\, then </math>0, a=b
 == Significance ==

2012-09-14T16:56:57Z

Theorems

← Older revision		Revision as of 12:56, 14 September 2012
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	<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F, a.(b+c)=a.b + a.c		<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F, a.(b+c)=a.b + a.c
	== Theorems ==		== Theorems ==

			'''Cancellation laws'''
			<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F

			if a+c=b+c, then a=b

			if ac=bc and c<math>\ne \!\,</math>0, a=b

	== Significance ==		== Significance ==

2012-09-14T16:49:04Z

← Older revision		Revision as of 12:49, 14 September 2012
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	<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F, a.(b+c)=a.b + a.c		<math>\forall \!\,</math> a, b, c <math>\in \!\,</math> F, a.(b+c)=a.b + a.c
			== Theorems ==

	== Significance ==		== Significance ==

2012-09-14T16:48:07Z

Definition of a field

← Older revision		Revision as of 12:48, 14 September 2012
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	'''F3:''' the existence of identity elements		'''F3:''' the existence of identity elements

	<math>\forall \!\,</math> a <math>\in \!\,</math>, a+o=a and a.1=a		<math>\forall \!\,</math> a <math>\in \!\,</math>, a+0=a and a.1=a

2012-09-14T16:47:51Z

Definition of a field

← Older revision		Revision as of 12:47, 14 September 2012
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	'''F3:''' the existence of identity elements		'''F3:''' the existence of identity elements

	<math>\forall \!\,</math> a <math>\in \!\,</math> ~~F\0~~, a+o=a and a.1=a		<math>\forall \!\,</math> a <math>\in \!\,</math>, a+o=a and a.1=a


	'''F4:''' existence of inverses		'''F4:''' existence of inverses

	<math>\forall \!\,</math> a <math>\in \!\,</math> F ,<math> \exists \!\,</math> c, d <math>\in \!\ </math> F such that a+c=o and a.d=1		<math>\forall \!\,</math> a <math>\in \!\,</math> F \0,<math> \exists \!\,</math> c, d <math>\in \!\ </math> F such that a+c=o and a.d=1