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
• 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 $\mathbb {R}$) along with two binary operations: + (plus) and · (times) and two special elements: 0 (zero) and 1 (one), such that the following laws hold true:
$\mathbb {R} 1:\forall a,b\in \mathbb {R} {\mbox{ s.th.}}\quad a+b=b+a\quad {\mbox{and}}\quad a\cdot b=b\cdot a$ (The Commutative Laws)
$\mathbb {R} 2:\forall a,b,c\in \mathbb {R} {\mbox{ s.th.}}\quad (a+b)+c=a+(b+c)\quad {\mbox{and}}\quad (a\cdot b)\cdot c=a\cdot (b\cdot c)$ (The Associative Laws)
$\mathbb {R} 3:0{\mbox{ is an additive unit}}\quad {\mbox{and}}\quad 1{\mbox{ is a multiplicative unit}}$ (The Existence of Units/Identities)
$\mathbb {R} 4:\forall a\in \mathbb {R} \ \exists b\in \mathbb {(} R){\mbox{ s.th.}}\ a+b=0$

This takes way too long. It is probably more practical to type the notes in Word and upload it onto the site, if we want a typesetted version of the notes.