Drorbn - User contributions [en]

14-240/Class Photo

2015-03-14T18:59:52Z

TSoDssy: Delete my information

Our class on September 24, 2014:

[[Image:14-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{14-240/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=

{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}



{{Photo Entry|last=Aleem|first=Asad|userid=AsadAleem|email=asad. aleem@ mail. utoronto. ca|location=Third from left in the seventh or eighth row, wearing a bright blue T-Shirt|comments=}}

{{Photo Entry|last=An|first=Ruiwen|userid=Christine An|email=christine. an@ mail. utoronto. ca|location=The "sunny girl" in dark brown (maybe) fifth from the left in the second row|comments=}}

{{Photo Entry|last=Cho|first=Alex|userid=Alex10002|email=aileefangirl. cho@ mail. utoronto. ca|location=Top left blue uoft sweather|comments=}}

{{Photo Entry|last=Doucet|first=Yannick|userid=Ydoucet|email=yannick. doucet@ mail. utoronto. ca|location=Second row, second from the right, with a cap on|comments=}}

{{Photo Entry|last=Field|first=Grace|userid=Grace.field|email=grace. field@ mail. utoronto. ca|location=The girl wearing a dark blue shirt in the third row, fifth from right|comments=}}

{{Photo Entry|last=Gomes|first=Andrew|userid=Agomes|email=andrew. gomes@ mail. utoronto. ca|location=The "young man" in the second row wearing a white T-shirt|comments=}}

{{Photo Entry|last=Guo|first=Chenyu|userid=chenyuguo|email=chenyu.guo@ mail. utoronto. ca|location=The "little girl" in the middle part of the last 4th row wearing a white shirt |comments=}}

{{Photo Entry|last=Huang|first=Charles|userid=Charlesh|email=cherls. huang@ mail. utoronto. ca|location=Fifth row, far left, stripped shirt|comments=}}

{{Photo Entry|last=Huang|first=Zane|userid=Bug|email=zane. huang@ mail.utoronto. ca|location=The dude way back in the eighth row on the right with his elbows up.|comments=}}

{{Photo Entry|last=Jiang|first=Yue|userid=Yue.Jiang|email=yuenj. jiang@ mail. utoronto. ca|location=The "little girl" in the second row, second from the left|comments=}}

{{Photo Entry|last=Lin|first=Tong|userid=Tong Lin|email=tong. lin@ mail. utoronto. ca|location=The gentleman third from the right in the seventh row|comments=}}

{{Photo Entry|last=Long|first=Austin|userid=AustinL|email=austin. long@ mail. utoronto. ca|location=The blonde male in the blue t-shirt near the left side stairs eighth row|comments=}}

{{Photo Entry|last=Looi|first=Shi Hao|userid=Shl|email=shihao. looi@ mail. utoronto. ca|location=Third row, third from left (if we are including the front row with just four or so chairs; otherwise, second row.) |comments=}}

{{Photo Entry|last=Luo|first=Danny (Xiao)|userid=Danny.luo|email=danny. luo (at) mail.utoronto. ca|location=The man third from the left in the second row, with a backpack in between his legs|comments=}}

{{Photo Entry|last=Mazin|first=Alexander|userid=AM|email=alexander.mazin@ mail.utoronto. ca|location=The man in the middle of the eight row blue t-shirt, arms folded|comments=}}

{{Photo Entry|last=McClure|first=Robertson|userid=r.mcclure|email=r. mcclure@ mail. toronto. edu|location= Sitting in the top left in a baby blue t-shirt wearing lanyard and small black necklace|comments=}}

{{Photo Entry|last=Morenz|first=Greg|userid=morenzg|email=greg.morenz@mail.utoronto.ca|location=Front Row, Third from the right, in all black.|comments=}}

{{Photo Entry|last=Nackan|first=Danny|userid=DannyN|email=danny. nackan@ mail. utoronto. ca|location=Second row, fourth from the right.|comments=}}

{{Photo Entry|last=Peng|first=Yuelin|userid=Ellen|email=yuelin.peng@ mail. utoronto. ca|location=The girl in a red shirt in the middle of the first row|comments=}}

{{Photo Entry|last=Piper|first=Cheyenne|userid=piperche|email=cheyenne.piper@ mail.utoronto. ca|location= 5th row,third from left,wearing a grey,red and black shirt,Behind the guy making the C sign,in front of the girl in white|comments= insert an inspirational quote here}}

{{Photo Entry|last=Shim|first=Soho|userid=Soho|email=soho. shim@ mail. utoronto. ca|location=The girl wearing a white T-shirt in the first row, third from the right|comments=}}

{{Photo Entry|last=Teunissen|first=Samuel|userid=Samuel Teunissen|email=samuel. teunissen@ mail. utoronto. ca|location= Fourth rom the right in the fourth row|comments=}}

{{Photo Entry|last=Tjandra|first=Donna|userid=Donna Tjandra|email=donna. tjandra@ mail. utoronto. ca|location=The girl in the 9th full row, 4th from the right, wearing a purple sweater|comments=}}

{{Photo Entry|last=Wei|first=Zhiyang|userid=Gianne|email=zhiyang. wei@ mail. utoronto. ca|location=second row, third from the right|comments=}}

{{Photo Entry|last=Wu|first=Boyang|userid=Boyang.wu|email=boyang. wu@mail. utoronto. ca|location=The boy with plaid shirt in the middle of fourth row from bottom and no neighbor sits beside him.|comments=}}

|}

14-240/Class Photo

2014-09-25T04:54:17Z

TSoDssy: /* Who We Are... */

14-240/Class Photo

2014-09-25T04:38:51Z

TSoDssy:

14-240/Classnotes for Monday September 15

2014-09-16T17:38:10Z

TSoDssy: more typesetting. Do we have the proof environment here?

Definition:
Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>.
Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>.

Theorem:

8. <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.
proof of 8: By F3 , <math>a \times 0 = a \times (0 + 0)</math>;
By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;
By F3 , <math>a \times 0 = 0 + a \times 0</math>;
By Thm P1,<math>0 = a \times 0</math>.

9. <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;
<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.
proof of 9: By F3 , <math>\times b = 0 \neq 1</math>.

10. <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.

11. <math>(-a) \times (-b) = a \times b</math>.

12. <math>a \times b = 0 \iff a = 0 or b = 0</math>.
proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a \times b = 0 \times b = 0</math>;
By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.
=> : Assume <math>a \times b = 0 </math> , if a = 0 we are done;
Otherwise , by P8 , <math>a \neq 0 </math> and we have <math>a \times b = 0 = a \times 0</math>;
by cancellation (P2) , <math>b = 0</math>.

<math>(a + b) \times (a - b) = a^2 - b^2</math>.
proof: By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))</math>
<math>= a \times a + a \times (-b) + b \times a + (-b) \times b</math>
<math>= a^2 - b^2</math>
Theorem :
<math>\exists! \iota : \Z \rightarrow F</math> s.t.
1. <math>\iota(0) = 0 , \iota(1) = 1</math>;
2. <math>\forall m ,n \in \Z, \iota(m+n) = \iota(m) + \iota(n)</math>;
3. <math>\forall m ,n \in \Z, \iota(m\times n) = \iota(m) \times \iota(n)</math>.

<math>\iota(2) = \iota(1+1) = \iota(1) + \iota(1) = 1 + 1;</math>
<math>\iota(3) = \iota(2+1) = \iota(2) + \iota(1) = \iota(2) + 1;</math>
......

In F2 , <math>27 ----> \iota(27) = \iota(26 + 1)</math>
<math>= \iota(26) + \iota(1)</math>
<math>= \iota(26) + 1</math>
<math>= \iota(13 \times 2) + 1</math>
<math>= \iota(2) \times \iota(13) + 1</math>
<math>= (1 + 1) \times \iota(13) + 1</math>
<math>= 0 \times \iota(13) + 1</math>
<math>= 1</math>

14-240/Classnotes for Monday September 15

2014-09-16T17:31:24Z

TSoDssy: fix typesetting, test latex

Definition:
Subtraction: if <math>a, b \in F, a - b = a + (-b)</math>.
Division: if <math>a, b \in F, a / b = a \times b^{-1}</math>.

Theorem:

8. <math>\forall a \in F</math>, <math>a \times 0 = 0</math>.
proof of 8: By F3 , <math>a \times 0 = a \times (0 + 0)</math>;
By F5 , <math>a \times (0 + 0) = a \times 0 + a \times 0</math>;
By F3 , <math>a \times 0 = 0 + a \times 0</math>;
By Thm P1,<math>0 = a \times 0</math>.

9. <math>\nexists b \in F</math> s.t. <math>0 \times b = 1</math>;
<math>\forall b \in F</math> s.t. <math>0 \times b \neq 1</math>.
proof of 9: By F3 , <math>\times b = 0 </math>is not equal to <math>1</math>.

10. <math>(-a) \times b = a \times (-b) = -(a \times b)</math>.

11. <math>(-a) \times (-b) = a \times b</math>.

12. <math>a \times b = 0 \iff a = 0 or b = 0</math>.
proof of 12: <= : By P8 , if <math>a = 0</math> , then <math>a \times b = 0 \times b = 0</math>;
By P8 , if <math>b = 0</math> , then <math>a \times b = a \times 0 = 0</math>.
=> : Assume <math>a \times b = 0 </math> , if a = 0 we are done;
Otherwise , by P8 , <math>a </math> is not equal to <math>0 </math>and we have <math>a \times b = 0 = a \times 0</math>;
by cancellation (P2) , <math>b = 0</math>.

<math>(a + b) \times (a - b) = a^2 - b^2</math>.
proof: By F5 , <math>(a + b) \times (a - b) = a \times (a + (-b)) + b \times (a + (-b))
= a \times a + a \times (-b) + b \times a + (-b) \times b
= a^2 - b^2</math>
Theorem :
<math>\exists! \iota : \Z \rightarrow F</math> s.t.
1. <math>\iota(0) = 0 , \iota(1) = 1</math>;
2. For every <math>m ,n \in Z</math> , <math>\iota(m+n) = \iota(m) + \iota(n)</math>;
3. For every <math>m ,n \in </math> , <math>\iota(m\times n) = \iota(m) \times \iota(n)</math>.

iota(2) = iota(1+1) = iota(1) + iota(1) = 1 + 1;
iota(3) = iota(2+1) = iota(2) + iota(1) = iota(2) + 1;
......

In F2 , <math>27 ----> iota(27) = iota(26 + 1)
= iota(26) + iota(1)
= iota(26) + 1
= iota(13 \times 2) + 1
= iota(2) \times iota(13) + 1
= (1 + 1) \times iota(13) + 1
= 0 \times iota(13) + 1
= 1</math>