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14-240/Classnotes for Monday September 15

2014-09-21T15:07:31Z

Penlong: /* Scanned notes */

{{14-240/Navigation}}
==Definition of Subtraction and Division==
* 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>.

==Basic Properties of a Field (cont'd)==

'''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</math> or <math>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^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>.

;Examples
<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>
\begin{align}
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
\end{align}
</math>

==Scanned notes==
http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2)
http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)

14-240/Classnotes for Monday September 15

2014-09-21T15:07:15Z

Penlong: /* Theorem */

{{14-240/Navigation}}
==Definition of Subtraction and Division==
* 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>.

==Basic Properties of a Field (cont'd)==

'''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</math> or <math>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^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>.

;Examples
<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>
\begin{align}
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
\end{align}
</math>

==Scanned notes===
http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2)
http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)

14-240/Classnotes for Monday September 15

2014-09-21T15:05:56Z

Penlong:

{{14-240/Navigation}}
==Definition of Subtraction and Division==
* 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>.

==Basic Properties of a Field (cont'd)==

'''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</math> or <math>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^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>.

;Examples
<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>
\begin{align}
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
\end{align}
</math>
http://drorbn.net/images/c/cd/MAT_240_lecture_3_%281_of_2%29.pdf (Lecture 3 notes by AM part 1 of 2)
http://drorbn.net/images/6/6a/MAT240_lectuire_3_%282_of_2%29.pdf (Lecture 3 notes by AM part 2 of 2)