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:
Edit1.jpg Edit2.jpg


The Real Numbers

The Real Numbers are a set (denoted by \mathbb{R}) along with two binary operations: + (plus) and · (times) and two special elements: 0 (zero) and 1 (one), such that the following laws hold true:

\mathbb{R}1: \forall a, b\in \mathbb{R} we have a+b=b+a and a\cdot b=b\cdot a (The Commutative Laws)

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

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

\mathbb{R}4: \forall a\in \mathbb{R} \ \exists b\in \mathbb{R} \mbox{ s.t.} \ a+b=0

This is incomplete.