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:
  • 06-240: Edit1.jpg
  • 06-240: Edit2.jpg

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

This is incomplete.