06-240/Classnotes For Tuesday, September 12
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- PDF notes by User:Harbansb: September 12 Notes.
- If I have made an error in my notes, or you would like the editable OpenOffice file, feel free to e-mail me at harbansb@msn.com.
- September 12 Notes for re-uploading, please email at jeff.matskin@utoronto.ca
- PDF notes by User:Alla: Week 1 Lecture 1 notes
- Below are a couple of lemmata critical to the derivation we did in class - the Professor left this little work to the students:
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.
