# 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 $\mathbb{R}$) 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.