06-240/Classnotes For Tuesday, September 12: Difference between revisions

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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 &middot; (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 &middot; (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.

Revision as of 17:09, 11 July 2007

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Notes

The Real Numbers

The Real Numbers are a set (denoted by [math]\displaystyle{ \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]\displaystyle{ \mathbb{R}1 }[/math]: [math]\displaystyle{ \forall a, b\in \mathbb{R} }[/math] we have [math]\displaystyle{ a+b=b+a }[/math] and [math]\displaystyle{ a\cdot b=b\cdot a }[/math] (The Commutative Laws)

[math]\displaystyle{ \mathbb{R}2 }[/math]: [math]\displaystyle{ \forall a, b, c\in \mathbb{R} }[/math] we have [math]\displaystyle{ (a+b)+c=a+(b+c) }[/math] and [math]\displaystyle{ (a\cdot b)\cdot c=a\cdot (b\cdot c) }[/math] (The Associative Laws)

[math]\displaystyle{ \mathbb{R}3 }[/math]: [math]\displaystyle{ 0 }[/math] is an additive unit and [math]\displaystyle{ 1 }[/math] is a multiplicative unit (The Existence of Units/Identities)

[math]\displaystyle{ \mathbb{R}4 }[/math]: [math]\displaystyle{ \forall a\in \mathbb{R} \ \exists b\in \mathbb{R} \mbox{ s.t.} \ a+b=0 }[/math]

This is incomplete.

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