14-240/Classnotes for Wednesday September 10 - Revision history

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

2014-12-08T05:56:44Z

Scanned Lecture Notes by AM

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	File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎\|page 2		File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎\|page 2
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			==Scanned Lecture Notes by [[User Boyang.wu\|Boyang.wu]]==

	[[File:w121.pdf]]		[[File:w121.pdf]]
	[[File:W122.pdf]]		[[File:W122.pdf]]

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

2014-12-08T05:18:59Z

Scanned Lecture Notes by AM

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	[[File:w121.pdf]]		[[File:w121.pdf]]
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Boyang.wu: /* Scanned Lecture Notes by AM */

2014-12-08T05:09:31Z

Scanned Lecture Notes by AM

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	File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎\|page 2		File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎\|page 2
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			[[File:w121.pdf]]

Boyang.wu at 05:02, 8 December 2014

2014-12-08T05:02:11Z

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	During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.		During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

	[[File:3.JPG]][[File:4。1.JPG]]

	===The Real Numbers===		===The Real Numbers===

Boyang.wu at 05:01, 8 December 2014

2014-12-08T05:01:27Z

← Older revision		Revision as of 01:01, 8 December 2014
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	During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.		During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

	[[File:3.JPG]]		[[File:3.JPG]][[File:4。1.JPG]]

	===The Real Numbers===		===The Real Numbers===

Boyang.wu at 04:59, 8 December 2014

2014-12-08T04:59:42Z

← Older revision		Revision as of 00:59, 8 December 2014
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	During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.		During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

	[[File:3.JPG]]~~===The Real Numbers===~~		[[File:3.JPG]]

			===The Real Numbers===

	====Properties of Real Numbers====		====Properties of Real Numbers====

Boyang.wu: /* The Real Numbers */

2014-12-08T04:57:42Z

The Real Numbers

← Older revision		Revision as of 00:57, 8 December 2014
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	During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.		During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

	===The Real Numbers===		[[File:3.JPG]]===The Real Numbers===

	====Properties of Real Numbers====		====Properties of Real Numbers====

Donna Tjandra: /* Basic Properties of Fields */

2014-10-19T20:23:25Z

Basic Properties of Fields

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	6. <math>\forall a, b, b' \in \mathbb{F}, a \ne 0 ~\&~ a * b = 1 = a * b' \Rightarrow b = b'</math>:		6. <math>\forall a, b, b' \in \mathbb{F}, a \ne 0 ~\&~ a * b = 1 = a * b' \Rightarrow b = b'</math>:
	In any field, if <math>a \ne 0</math>, "<math>a^{-1}</math>" makes sense.		In any field, if <math>a \ne 0</math>, "<math>a^{-1}</math>" makes sense.


	Proof of 5 :		Proof of 5 :
	1. <math>a + b = 0 = a + b'</math>.		1. <math>a + b = 0 = a + b'</math>.

Yue.Jiang at 21:04, 23 September 2014

2014-09-23T21:04:11Z

Donna Tjandra: /* Properties of Real Numbers */

2014-09-20T22:14:41Z

Properties of Real Numbers

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	R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)		R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

	R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the ~~"additive unit" and 1 as the "multiplicative unit")~~		R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the
			"additive unit" and 1 as the "multiplicative unit")

	R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;		R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;

← Older revision		Revision as of 17:04, 23 September 2014
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	such that the following properties hold.		such that the following properties hold.

	R1 : <math>\forall a, b \in \R, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)		* R1 : <math>\forall a, b \in \R, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

	R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)		* R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

	R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the		* R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the
	"additive unit" and 1 as the "multiplicative unit")		\| "additive unit" and 1 as the "multiplicative unit")

	R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;		* R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;
	<math>\forall a \in \R, a \ne 0 \Rightarrow \exists b \in \R, a * b = 1</math> (existence of inverses)		\| <math>\forall a \in \R, a \ne 0 \Rightarrow \exists b \in \R, a * b = 1</math> (existence of inverses)

	R5 : <math>\forall a, b, c \in \R, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)		* R5 : <math>\forall a, b, c \in \R, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)

	====Properties That Do Not Follow from R1-R5====		====Properties That Do Not Follow from R1-R5====
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	and along with a pair <math>(0, 1) \in \mathbb{F}, 0 \ne 1</math>, such that the following properties hold.		and along with a pair <math>(0, 1) \in \mathbb{F}, 0 \ne 1</math>, such that the following properties hold.

	F1 : <math>\forall a, b \in \mathbb{F}, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)		* F1 : <math>\forall a, b \in \mathbb{F}, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

	F2 : <math>\forall a, b, c \in \mathbb{F}, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)		* F2 : <math>\forall a, b, c \in \mathbb{F}, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

	F3 : <math>\forall a \in \mathbb{F}, a + 0 = a ~\&~ a * 1 = a</math> (existence of units)		* F3 : <math>\forall a \in \mathbb{F}, a + 0 = a ~\&~ a * 1 = a</math> (existence of units)

	F4 : <math>\forall a \in \mathbb{F}, \exists b \in \mathbb{F}, a + b = 0</math>;		* F4 : <math>\forall a \in \mathbb{F}, \exists b \in \mathbb{F}, a + b = 0</math>;
	<math>\forall a \in \mathbb{F}, a \ne 0 \Rightarrow \exists b \in \mathbb{F}, a * b = 1</math> (existence of inverses)		\| <math>\forall a \in \mathbb{F}, a \ne 0 \Rightarrow \exists b \in \mathbb{F}, a * b = 1</math> (existence of inverses)

	F5 : <math>\forall a, b, c \in \mathbb{F}, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)		* F5 : <math>\forall a, b, c \in \mathbb{F}, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)
	====Examples====		====Examples====