Drorbn - User contributions [en]

14-240/Classnotes for Wednesday September 10

2014-10-19T20:23:25Z

Donna Tjandra: /* Basic Properties of Fields */

{{14-240/Navigation}}
Knowledge about Fields:

During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

===The Real Numbers===

====Properties of Real Numbers====

The real numbers are a set <math>\R</math> with two binary operations:

<math>+ : \R \times \R \rightarrow \R</math>

<math>* : \R \times \R \rightarrow \R</math>

such that the following properties hold.

* R1 : <math>\forall a, b \in \R, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

* R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

* R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the
| "additive unit" and 1 as the "multiplicative unit")

* R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;
| <math>\forall a \in \R, a \ne 0 \Rightarrow \exists b \in \R, a * b = 1</math> (existence of inverses)

* R5 : <math>\forall a, b, c \in \R, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)

====Properties That Do Not Follow from R1-R5====

There are properties which are true for <math>\R</math>, but do not follow from R1 to R5. For example ('''note''' that OR in mathematics denotes an "inclusive or"):
<math>\forall a \in \R, \exists x \in \R, a = x^2</math> OR <math>-a = x^2</math> (the existence of square roots)

Consider another set that satisfies all the properties R1 to R5. In <math>\Q</math> (the rational numbers), let us take </math>a = 2</math>. There is no <math>x \in \Q</math> such that <math>x^2 = a = 2</math>, so the statement above is not true for the rational numbers!

-------------------------------------------------------------------------------------------------------------------------------------------------------

===Fields===

====Definition====

A "field" is a set <math>\mathbb{F}</math> along with a pair of binary operations:
<math>+ : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

<math>* : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

and along with a pair <math>(0, 1) \in \mathbb{F}, 0 \ne 1</math>, such that the following properties hold.

* F1 : <math>\forall a, b \in \mathbb{F}, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

* F2 : <math>\forall a, b, c \in \mathbb{F}, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

* F3 : <math>\forall a \in \mathbb{F}, a + 0 = a ~\&~ a * 1 = a</math> (existence of units)

* F4 : <math>\forall a \in \mathbb{F}, \exists b \in \mathbb{F}, a + b = 0</math>;
| <math>\forall a \in \mathbb{F}, a \ne 0 \Rightarrow \exists b \in \mathbb{F}, a * b = 1</math> (existence of inverses)

* F5 : <math>\forall a, b, c \in \mathbb{F}, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)
====Examples====

# <math>\R</math> is a field.
# <math>\Q</math> (the rational numbers) is a field.
# <math>\C</math> (the complex numbers) is a field.
# <math>\mathbb{F} = \{0, 1\}</math> with operations defined as follows (known as <math>\mathbb{F}_2</math> or <math>\Z/2</math>) is a field:

{| class="wikitable"
|-
! +
! 0
! 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 0
|}

{| class="wikitable"
|-
! *
! 0
! 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

 

More generally, for every prime number <math>P</math>, <math>\mathbb{F}_p = \{0, 1, 2, 3, \cdots, p - 1\}</math> is a field, with operations defined by
<math>(a, b) \rightarrow a + b \mod P</math>.

An example: <math>\mathbb{F}_7 = \{0, 1, 2, 3, 4, 5, 6\}</math>, the operations are like remainders when divided by 7, or "like remainders mod 7". For example, <math>4 + 6 = 4 + 6 \mod 7</math> and <math>3 * 5 = 3 * 5 \mod 7</math>.

====Basic Properties of Fields====

'''Theorem''':
Let <math>\mathbb{F}</math> be a field, and let <math>a, b, c</math> denote elements of <math>\mathbb{F}</math>. Then:
# <math>a + b = c + b \Rightarrow a = c</math> (cancellation law)
# <math>b \ne 0 ~\&~ a * b = c * b \Rightarrow a = c</math>
Proof of 1:
1. By F4, <math>\exists b' \in \mathbb{F}, b + b' = 0</math>.
We know that <math>a + b = c + b</math>;
Therefore <math>(a + b) + b' = (c + b) + b'</math>.
2. By F2, <math>a + (b + b') = c + (b + b')</math>,
so by the choice of <math>b'</math>, we know that <math>a + 0 = c + 0</math>.
3. Therefore, by F3, <math>a = c</math>.
＾_＾

Proof of 2: more or less the same.

3. If <math>0' \in \mathbb{F}</math> is "like 0", then it is 0:
If <math>0' \in \mathbb{F}</math> satisfies <math>\forall a \in \mathbb{F}, a + 0' = a</math>, then 0' = 0.

4. If <math>1' \in \mathbb{F}</math> is "like 1", then it is 1:
If <math>1' \in \mathbb{F}</math> satisfies that <math>\forall a \in \mathbb{F}, a * 1' = a</math>, then 1' = 1.

Proof of 3 :
1. By F3 , 0' = 0' + 0.
2. By F1 , 0' + 0 = 0 + 0'.
3. By assumption on 0', 0' = 0 + 0' = 0.
＾_＾

5. <math>\forall a, b, b' \in \mathbb{F}, a + b = 0 ~\&~ a + b' = 0 \Rightarrow b = b'</math>:
In any field "<math>-a</math>" makes sense because it is unique -- it has an unambiguous meaning.
<math>(-a):</math> the <math>b</math> such that <math>a + b = 0</math>.

6. <math>\forall a, b, b' \in \mathbb{F}, a \ne 0 ~\&~ a * b = 1 = a * b' \Rightarrow b = b'</math>:
In any field, if <math>a \ne 0</math>, "<math>a^{-1}</math>" makes sense.

Proof of 5 :
1. <math>a + b = 0 = a + b'</math>.
2. By F1, <math>b + a = b'+ a</math>.
3. By cancellation, <math>b = b'</math>.
＾_＾

7. <math>-(-a) = a</math> and when <math>a \ne 0</math>, <math>(a^{-1})^{-1} = a</math>.

Proof of 7 :
1. By definition, <math>a + (-a) = 0</math>. (*)
2. By definition, <math>(-a) + (-(-a) = 0</math>.
3. By (*) and F1, <math>(-a) + a = 0</math>.
4. By property 5, <math>-(-a) = a</math>.
＾_＾

===Scanned Lecture Notes by [[User:AM|AM]]===

<gallery>
File:MAT 240 (1 of 2) Sept 10, 2014.pdf‎|page 1
File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎|page 2
</gallery>

14-240/Classnotes for Monday October 6

2014-10-10T01:48:23Z

Donna Tjandra:

{{14-240/Navigation}}
*http://drorbn.net/images/d/d3/MAT240_%28Oct_6%2C_2014%29_1_of_2.pdf (Notes by AM 1 of 2)
*http://drorbn.net/images/7/7c/MAT240_%28Oct_6%2C_2014%29_2_of_2.pdf (Notes by AM 2 of 2)

14-240/Classnotes for Wednesday October 8

2014-10-09T17:25:22Z

Donna Tjandra:

{{14-240/Navigation}}
==Scanned Lecture Notes by [[User Yue.Jiang|Yue Jiang]]==

<gallery>
File:October 8 note 1.jpeg|page 1
File:October 8 note 2.jpeg|page 2
File:October 8 note 3.jpeg|page 3
File:4.jpg|page 4
File:5.jpg|page 5
</gallery>

14-240/Class Photo

2014-09-28T20:34:13Z

Donna Tjandra:

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=0
|-
!First Name
!Last Name
!ID wcashore
!e-mail
!Location
!Comments

{{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=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=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=Huang|first=Charles|userid=Charlesh|email=cherls.huang@ mail. utoronto. ca|location=Fifth row, far left, stripped shirt|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=Luo|first=Danny (Xiao)|userid=Danny.luo|email=danny.luo@ 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=Shen|first=Crystal|userid=tsodssy|email=crystal.shen@ mail. utoronto. ca|location=The girl not facing the camera (How did this happen?) in the front row right corner|comments=}}

{{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=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=}}

|}

14-240/Classnotes for Wednesday September 24

2014-09-26T02:34:30Z

Donna Tjandra:

{{14-240/Navigation}}
http://drorbn.net/images/0/06/MAT240_Sept_22_%281_of_4%29.pdf (Class Notes by AM 1 of 4)
http://drorbn.net/images/7/78/MAT240_Sept_24_%282_of_4%29.pdf (Class Notes by AM 2 of 4)
http://drorbn.net/images/4/46/MAT240_Sept_24_%283_of4%29.pdf (Class Notes by AM 3 of 4)
http://drorbn.net/images/8/89/MAT240_Sept_24_%284_of_4%29.pdf (Class Notes by AM 4 of 4)

14-240/Classnotes for Monday September 22

2014-09-25T03:25:36Z

Donna Tjandra:

{{14-240/Navigation}}
Polar coordinates:
* <math>r \times e^{i\theta} = r \times cos\theta + i \times rsin\theta</math>
* <math>r_1 \times e^{i\theta_2} = r_1 \times (cos\theta + sin\theta</math>

The Fundamantal Theorem of Algebra:
<math>a_n \times z^{n} + a_n-1 \times z^{n-1} + \dots + a_0</math>
where <math>a_i \in C </math>and<math> a_i != 0</math> has a soluion <math>z \in C</math>
In particular, <math>z^{2} - 1 = 0</math> has a solution.

* Forces can multiple by a "scalar"(number).
No "multiplication" of forces.

Definition of Vector Space:
A "Vector Space" over a field F is a set V with a special element <math>O_v \in V</math> and two binary operations:
* <math>+ : V \times V -> V</math>
* <math>\times : V \times V -> V</math>

s.t.
* <math>VS_1 : \forall x, y \in V, x + y = y + x</math>.
* <math>VS_2 : \forall x, y, z \in V, x + (y + z) = (x + y) + z</math>.
* <math>VS_3 : \forall x \in V, x + 0 = x</math>.
* <math>VS_4 : \forall x \in V, \exists y \in V, x + y = 0</math>.
* <math>VS_5 : \forall x \in V, 1 \times x = x</math>.
* <math>VS_6 : \forall a, b \in F, \forall x \in V, a(bx) = (ab)x</math>.
* <math>VS_7 : \forall a \in F, \forall x, y \in V, a(x + y) = ax + ay</math>.
* <math>VS_8 : \forall a, b \in F, \forall x \in V, (a + b)x = ax + bx</math>.

14-240/Classnotes for Wednesday September 10

2014-09-20T22:14:41Z

Donna Tjandra: /* Properties of Real Numbers */

{{14-240/Navigation}}
Knowledge about Fields:

During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

===The Real Numbers===

====Properties of Real Numbers====

The real numbers are a set <math>\R</math> with two binary operations:

<math>+ : \R \times \R \rightarrow \R</math>

<math>* : \R \times \R \rightarrow \R</math>

such that the following properties hold.

R1 : <math>\forall a, b \in \R, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the
"additive unit" and 1 as the "multiplicative unit")

R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;
<math>\forall a \in \R, a \ne 0 \Rightarrow \exists b \in \R, a * b = 1</math> (existence of inverses)

R5 : <math>\forall a, b, c \in \R, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)

====Properties That Do Not Follow from R1-R5====

There are properties which are true for <math>\R</math>, but do not follow from R1 to R5. For example ('''note''' that OR in mathematics denotes an "inclusive or"):
<math>\forall a \in \R, \exists x \in \R, a = x^2</math> OR <math>-a = x^2</math> (the existence of square roots)

Consider another set that satisfies all the properties R1 to R5. In <math>\Q</math> (the rational numbers), let us take </math>a = 2</math>. There is no <math>x \in \Q</math> such that <math>x^2 = a = 2</math>, so the statement above is not true for the rational numbers!

-------------------------------------------------------------------------------------------------------------------------------------------------------

===Fields===

====Definition====

A "field" is a set <math>\mathbb{F}</math> along with a pair of binary operations:
<math>+ : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

<math>* : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

and along with a pair <math>(0, 1) \in \mathbb{F}, 0 \ne 1</math>, such that the following properties hold.

F1 : <math>\forall a, b \in \mathbb{F}, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

F2 : <math>\forall a, b, c \in \mathbb{F}, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

F3 : <math>\forall a \in \mathbb{F}, a + 0 = a ~\&~ a * 1 = a</math> (existence of units)

F4 : <math>\forall a \in \mathbb{F}, \exists b \in \mathbb{F}, a + b = 0</math>;
<math>\forall a \in \mathbb{F}, a \ne 0 \Rightarrow \exists b \in \mathbb{F}, a * b = 1</math> (existence of inverses)

F5 : <math>\forall a, b, c \in \mathbb{F}, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)
====Examples====

# <math>\R</math> is a field.
# <math>\Q</math> (the rational numbers) is a field.
# <math>\C</math> (the complex numbers) is a field.
# <math>\mathbb{F} = \{0, 1\}</math> with operations defined as follows (known as <math>\mathbb{F}_2</math> or <math>\Z/2</math>) is a field:

{| class="wikitable"
|-
! +
! 0
! 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 0
|}

{| class="wikitable"
|-
! *
! 0
! 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

 

More generally, for every prime number <math>P</math>, <math>\mathbb{F}_p = \{0, 1, 2, 3, \cdots, p - 1\}</math> is a field, with operations defined by
<math>(a, b) \rightarrow a + b \mod P</math>.

An example: <math>\mathbb{F}_7 = \{0, 1, 2, 3, 4, 5, 6\}</math>, the operations are like remainders when divided by 7, or "like remainders mod 7". For example, <math>4 + 6 = 4 + 6 \mod 7</math> and <math>3 * 5 = 3 * 5 \mod 7</math>.

====Basic Properties of Fields====

'''Theorem''':
Let <math>\mathbb{F}</math> be a field, and let <math>a, b, c</math> denote elements of <math>\mathbb{F}</math>. Then:
# <math>a + b = c + b \Rightarrow a = c</math> (cancellation law)
# <math>b \ne 0 ~\&~ a * b = c * b \Rightarrow a = c</math>
Proof of 1:
1. By F4, <math>\exists b' \in \mathbb{F}, b + b' = 0</math>.
We know that <math>a + b = c + b</math>;
Therefore <math>(a + b) + b' = (c + b) + b'</math>.
2. By F2, <math>a + (b + b') = c + (b + b')</math>,
so by the choice of <math>b'</math>, we know that <math>a + 0 = c + 0</math>.
3. Therefore, by F3, <math>a = c</math>.
＾_＾

Proof of 2: more or less the same.

3. If <math>0' \in \mathbb{F}</math> is "like 0", then it is 0:
If <math>0' \in \mathbb{F}</math> satisfies <math>\forall a \in \mathbb{F}, a + 0' = a</math>, then 0' = 0.

4. If <math>1' \in \mathbb{F}</math> is "like 1", then it is 1:
If <math>1' \in \mathbb{F}</math> satisfies that <math>\forall a \in \mathbb{F}, a * 1' = a</math>, then 1' = 1.

Proof of 3 :
1. By F3 , 0' = 0' + 0.
2. By F1 , 0' + 0 = 0 + 0'.
3. By assumption on 0', 0' = 0 + 0' = 0.
＾_＾

5. <math>\forall a, b, b' \in \mathbb{F}, a + b = 0 ~\&~ a + b' = 0 \Rightarrow b = b'</math>:
In any field "<math>-a</math>" makes sense because it is unique -- it has an unambiguous meaning.
<math>(-a):</math> the <math>b</math> such that <math>a + b = 0</math>.

6. <math>\forall a, b, b' \in \mathbb{F}, a \ne 0 ~\&~ a * b = 1 = a * b' \Rightarrow b = b'</math>:
In any field, if <math>a \ne 0</math>, "<math>a^{-1}</math>" makes sense.

Proof of 5 :
1. <math>a + b = 0 = a + b'</math>.
2. By F1, <math>b + a = b'+ a</math>.
3. By cancellation, <math>b = b'</math>.
＾_＾

7. <math>-(-a) = a</math> and when <math>a \ne 0</math>, <math>(a^{-1})^{-1} = a</math>.

Proof of 7 :
1. By definition, <math>a + (-a) = 0</math>. (*)
2. By definition, <math>(-a) + (-(-a) = 0</math>.
3. By (*) and F1, <math>(-a) + a = 0</math>.
4. By property 5, <math>-(-a) = a</math>.
＾_＾

===Scanned Lecture Notes by [[User:AM|AM]]===

<gallery>
File:MAT 240 (1 of 2) Sept 10, 2014.pdf‎|page 1
File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎|page 2
</gallery>

14-240/Classnotes for Wednesday September 10

2014-09-20T22:08:01Z

Donna Tjandra: /* Properties of Real Numbers */

{{14-240/Navigation}}
Knowledge about Fields:

During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

===The Real Numbers===

====Properties of Real Numbers====

The real numbers are a set <math>\R</math> with two binary operations:

<math>+ : \R \times \R \rightarrow \R</math>

<math>* : \R \times \R \rightarrow \R</math>

such that the following properties hold.

R1 : <math>\forall a, b \in \R, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the "additive unit" and 1 as the "multiplicative unit")

R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;
<math>\forall a \in \R, a \ne 0 \Rightarrow \exists b \in \R, a * b = 1</math> (existence of inverses)

R5 : <math>\forall a, b, c \in \R, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)

====Properties That Do Not Follow from R1-R5====

There are properties which are true for <math>\R</math>, but do not follow from R1 to R5. For example ('''note''' that OR in mathematics denotes an "inclusive or"):
<math>\forall a \in \R, \exists x \in \R, a = x^2</math> OR <math>-a = x^2</math> (the existence of square roots)

Consider another set that satisfies all the properties R1 to R5. In <math>\Q</math> (the rational numbers), let us take </math>a = 2</math>. There is no <math>x \in \Q</math> such that <math>x^2 = a = 2</math>, so the statement above is not true for the rational numbers!

-------------------------------------------------------------------------------------------------------------------------------------------------------

===Fields===

====Definition====

A "field" is a set <math>\mathbb{F}</math> along with a pair of binary operations:
<math>+ : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

<math>* : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

and along with a pair <math>(0, 1) \in \mathbb{F}, 0 \ne 1</math>, such that the following properties hold.

F1 : <math>\forall a, b \in \mathbb{F}, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

F2 : <math>\forall a, b, c \in \mathbb{F}, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

F3 : <math>\forall a \in \mathbb{F}, a + 0 = a ~\&~ a * 1 = a</math> (existence of units)

F4 : <math>\forall a \in \mathbb{F}, \exists b \in \mathbb{F}, a + b = 0</math>;
<math>\forall a \in \mathbb{F}, a \ne 0 \Rightarrow \exists b \in \mathbb{F}, a * b = 1</math> (existence of inverses)

F5 : <math>\forall a, b, c \in \mathbb{F}, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)
====Examples====

# <math>\R</math> is a field.
# <math>\Q</math> (the rational numbers) is a field.
# <math>\C</math> (the complex numbers) is a field.
# <math>\mathbb{F} = \{0, 1\}</math> with operations defined as follows (known as <math>\mathbb{F}_2</math> or <math>\Z/2</math>) is a field:

{| class="wikitable"
|-
! +
! 0
! 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 0
|}

{| class="wikitable"
|-
! *
! 0
! 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

 

More generally, for every prime number <math>P</math>, <math>\mathbb{F}_p = \{0, 1, 2, 3, \cdots, p - 1\}</math> is a field, with operations defined by
<math>(a, b) \rightarrow a + b \mod P</math>.

An example: <math>\mathbb{F}_7 = \{0, 1, 2, 3, 4, 5, 6\}</math>, the operations are like remainders when divided by 7, or "like remainders mod 7". For example, <math>4 + 6 = 4 + 6 \mod 7</math> and <math>3 * 5 = 3 * 5 \mod 7</math>.

====Basic Properties of Fields====

'''Theorem''':
Let <math>\mathbb{F}</math> be a field, and let <math>a, b, c</math> denote elements of <math>\mathbb{F}</math>. Then:
# <math>a + b = c + b \Rightarrow a = c</math> (cancellation law)
# <math>b \ne 0 ~\&~ a * b = c * b \Rightarrow a = c</math>
Proof of 1:
1. By F4, <math>\exists b' \in \mathbb{F}, b + b' = 0</math>.
We know that <math>a + b = c + b</math>;
Therefore <math>(a + b) + b' = (c + b) + b'</math>.
2. By F2, <math>a + (b + b') = c + (b + b')</math>,
so by the choice of <math>b'</math>, we know that <math>a + 0 = c + 0</math>.
3. Therefore, by F3, <math>a = c</math>.
＾_＾

Proof of 2: more or less the same.

3. If <math>0' \in \mathbb{F}</math> is "like 0", then it is 0:
If <math>0' \in \mathbb{F}</math> satisfies <math>\forall a \in \mathbb{F}, a + 0' = a</math>, then 0' = 0.

4. If <math>1' \in \mathbb{F}</math> is "like 1", then it is 1:
If <math>1' \in \mathbb{F}</math> satisfies that <math>\forall a \in \mathbb{F}, a * 1' = a</math>, then 1' = 1.

Proof of 3 :
1. By F3 , 0' = 0' + 0.
2. By F1 , 0' + 0 = 0 + 0'.
3. By assumption on 0', 0' = 0 + 0' = 0.
＾_＾

5. <math>\forall a, b, b' \in \mathbb{F}, a + b = 0 ~\&~ a + b' = 0 \Rightarrow b = b'</math>:
In any field "<math>-a</math>" makes sense because it is unique -- it has an unambiguous meaning.
<math>(-a):</math> the <math>b</math> such that <math>a + b = 0</math>.

6. <math>\forall a, b, b' \in \mathbb{F}, a \ne 0 ~\&~ a * b = 1 = a * b' \Rightarrow b = b'</math>:
In any field, if <math>a \ne 0</math>, "<math>a^{-1}</math>" makes sense.

Proof of 5 :
1. <math>a + b = 0 = a + b'</math>.
2. By F1, <math>b + a = b'+ a</math>.
3. By cancellation, <math>b = b'</math>.
＾_＾

7. <math>-(-a) = a</math> and when <math>a \ne 0</math>, <math>(a^{-1})^{-1} = a</math>.

Proof of 7 :
1. By definition, <math>a + (-a) = 0</math>. (*)
2. By definition, <math>(-a) + (-(-a) = 0</math>.
3. By (*) and F1, <math>(-a) + a = 0</math>.
4. By property 5, <math>-(-a) = a</math>.
＾_＾

===Scanned Lecture Notes by [[User:AM|AM]]===

<gallery>
File:MAT 240 (1 of 2) Sept 10, 2014.pdf‎|page 1
File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎|page 2
</gallery>

14-240/Classnotes for Wednesday September 10

2014-09-20T22:07:21Z

Donna Tjandra: /* Properties of Real Numbers */

{{14-240/Navigation}}
Knowledge about Fields:

During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

===The Real Numbers===

====Properties of Real Numbers====

The real numbers are a set <math>\R</math> with two binary operations:

<math>+ : \R \times \R \rightarrow \R</math>

<math>* : \R \times \R \rightarrow \R</math>

such that the following properties hold.

R1 : <math>\forall a, b \in \R, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the "additive unit"
and 1 as the "multiplicative unit")

R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;
<math>\forall a \in \R, a \ne 0 \Rightarrow \exists b \in \R, a * b = 1</math> (existence of inverses)

R5 : <math>\forall a, b, c \in \R, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)

====Properties That Do Not Follow from R1-R5====

There are properties which are true for <math>\R</math>, but do not follow from R1 to R5. For example ('''note''' that OR in mathematics denotes an "inclusive or"):
<math>\forall a \in \R, \exists x \in \R, a = x^2</math> OR <math>-a = x^2</math> (the existence of square roots)

Consider another set that satisfies all the properties R1 to R5. In <math>\Q</math> (the rational numbers), let us take </math>a = 2</math>. There is no <math>x \in \Q</math> such that <math>x^2 = a = 2</math>, so the statement above is not true for the rational numbers!

-------------------------------------------------------------------------------------------------------------------------------------------------------

===Fields===

====Definition====

A "field" is a set <math>\mathbb{F}</math> along with a pair of binary operations:
<math>+ : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

<math>* : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

and along with a pair <math>(0, 1) \in \mathbb{F}, 0 \ne 1</math>, such that the following properties hold.

F1 : <math>\forall a, b \in \mathbb{F}, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

F2 : <math>\forall a, b, c \in \mathbb{F}, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

F3 : <math>\forall a \in \mathbb{F}, a + 0 = a ~\&~ a * 1 = a</math> (existence of units)

F4 : <math>\forall a \in \mathbb{F}, \exists b \in \mathbb{F}, a + b = 0</math>;
<math>\forall a \in \mathbb{F}, a \ne 0 \Rightarrow \exists b \in \mathbb{F}, a * b = 1</math> (existence of inverses)

F5 : <math>\forall a, b, c \in \mathbb{F}, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)
====Examples====

# <math>\R</math> is a field.
# <math>\Q</math> (the rational numbers) is a field.
# <math>\C</math> (the complex numbers) is a field.
# <math>\mathbb{F} = \{0, 1\}</math> with operations defined as follows (known as <math>\mathbb{F}_2</math> or <math>\Z/2</math>) is a field:

{| class="wikitable"
|-
! +
! 0
! 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 0
|}

{| class="wikitable"
|-
! *
! 0
! 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

 

More generally, for every prime number <math>P</math>, <math>\mathbb{F}_p = \{0, 1, 2, 3, \cdots, p - 1\}</math> is a field, with operations defined by
<math>(a, b) \rightarrow a + b \mod P</math>.

An example: <math>\mathbb{F}_7 = \{0, 1, 2, 3, 4, 5, 6\}</math>, the operations are like remainders when divided by 7, or "like remainders mod 7". For example, <math>4 + 6 = 4 + 6 \mod 7</math> and <math>3 * 5 = 3 * 5 \mod 7</math>.

====Basic Properties of Fields====

'''Theorem''':
Let <math>\mathbb{F}</math> be a field, and let <math>a, b, c</math> denote elements of <math>\mathbb{F}</math>. Then:
# <math>a + b = c + b \Rightarrow a = c</math> (cancellation law)
# <math>b \ne 0 ~\&~ a * b = c * b \Rightarrow a = c</math>
Proof of 1:
1. By F4, <math>\exists b' \in \mathbb{F}, b + b' = 0</math>.
We know that <math>a + b = c + b</math>;
Therefore <math>(a + b) + b' = (c + b) + b'</math>.
2. By F2, <math>a + (b + b') = c + (b + b')</math>,
so by the choice of <math>b'</math>, we know that <math>a + 0 = c + 0</math>.
3. Therefore, by F3, <math>a = c</math>.
＾_＾

Proof of 2: more or less the same.

3. If <math>0' \in \mathbb{F}</math> is "like 0", then it is 0:
If <math>0' \in \mathbb{F}</math> satisfies <math>\forall a \in \mathbb{F}, a + 0' = a</math>, then 0' = 0.

4. If <math>1' \in \mathbb{F}</math> is "like 1", then it is 1:
If <math>1' \in \mathbb{F}</math> satisfies that <math>\forall a \in \mathbb{F}, a * 1' = a</math>, then 1' = 1.

Proof of 3 :
1. By F3 , 0' = 0' + 0.
2. By F1 , 0' + 0 = 0 + 0'.
3. By assumption on 0', 0' = 0 + 0' = 0.
＾_＾

5. <math>\forall a, b, b' \in \mathbb{F}, a + b = 0 ~\&~ a + b' = 0 \Rightarrow b = b'</math>:
In any field "<math>-a</math>" makes sense because it is unique -- it has an unambiguous meaning.
<math>(-a):</math> the <math>b</math> such that <math>a + b = 0</math>.

6. <math>\forall a, b, b' \in \mathbb{F}, a \ne 0 ~\&~ a * b = 1 = a * b' \Rightarrow b = b'</math>:
In any field, if <math>a \ne 0</math>, "<math>a^{-1}</math>" makes sense.

Proof of 5 :
1. <math>a + b = 0 = a + b'</math>.
2. By F1, <math>b + a = b'+ a</math>.
3. By cancellation, <math>b = b'</math>.
＾_＾

7. <math>-(-a) = a</math> and when <math>a \ne 0</math>, <math>(a^{-1})^{-1} = a</math>.

Proof of 7 :
1. By definition, <math>a + (-a) = 0</math>. (*)
2. By definition, <math>(-a) + (-(-a) = 0</math>.
3. By (*) and F1, <math>(-a) + a = 0</math>.
4. By property 5, <math>-(-a) = a</math>.
＾_＾

===Scanned Lecture Notes by [[User:AM|AM]]===

<gallery>
File:MAT 240 (1 of 2) Sept 10, 2014.pdf‎|page 1
File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎|page 2
</gallery>

14-240/Classnotes for Wednesday September 10

2014-09-14T23:48:28Z

Donna Tjandra: /* Basic Properties of Fields */

Knowledge about Fields:

During this lecture, we first talked about the properties of the real numbers. Then we applied these properties to the "Field". At the end of the lecture, we learned how to prove basic properties of fields.

===The Real Numbers===

====Properties of Real Numbers====

The real numbers are a set <math>\R</math> with two binary operations:

<math>+ : \R \times \R \rightarrow \R</math>

<math>* : \R \times \R \rightarrow \R</math>

such that the following properties hold.

R1 : <math>\forall a, b \in \R, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

R2 : <math>\forall a, b, c \in \R, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

R3 : <math>\forall a \in \R, a + 0 = a ~\&~ a * 1 = a</math> (existence of units: 0 is known as the "additive unit" and 1 as the "multiplicative unit")

R4 : <math>\forall a \in \R, \exists b \in \R, a + b = 0</math>;
<math>\forall a \in \R, a \ne 0 \Rightarrow \exists b \in \R, a * b = 1</math> (existence of inverses)

R5 : <math>\forall a, b, c \in \R, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)

====Properties That Do Not Follow from R1-R5====

There are properties which are true for <math>\R</math>, but do not follow from R1 to R5. For example ('''note''' that OR in mathematics denotes an "inclusive or"):
<math>\forall a \in \R, \exists x \in \R, a = x^2</math> OR <math>-a = x^2</math> (the existence of square roots)

Consider another set that satisfies all the properties R1 to R5. In <math>\Q</math> (the rational numbers), let us take </math>a = 2</math>. There is no <math>x \in \Q</math> such that <math>x^2 = a = 2</math>, so the statement above is not true for the rational numbers!

-------------------------------------------------------------------------------------------------------------------------------------------------------

===Fields===

====Definition====

A "field" is a set <math>\mathbb{F}</math> along with a pair of binary operations:
<math>+ : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

<math>* : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}</math>

and along with a pair <math>(0, 1) \in \mathbb{F}, 0 \ne 1</math>, such that the following properties hold.

F1 : <math>\forall a, b \in \mathbb{F}, a + b = b + a ~\&~ a * b = b * a</math> (the commutative law)

F2 : <math>\forall a, b, c \in \mathbb{F}, (a + b) + c = a + (b + c) ~\&~ (a * b) * c = a * (b * c)</math> (the associative law)

F3 : <math>\forall a \in \mathbb{F}, a + 0 = a ~\&~ a * 1 = a</math> (existence of units)

F4 : <math>\forall a \in \mathbb{F}, \exists b \in \mathbb{F}, a + b = 0</math>;
<math>\forall a \in \mathbb{F}, a \ne 0 \Rightarrow \exists b \in \mathbb{F}, a * b = 1</math> (existence of inverses)

F5 : <math>\forall a, b, c \in \mathbb{F}, (a + b) * c = (a * c) + (b * c)</math> (the distributive law)
====Examples====

# <math>\R</math> is a field.
# <math>\Q</math> (the rational numbers) is a field.
# <math>\C</math> (the complex numbers) is a field.
# <math>\mathbb{F} = \{0, 1\}</math> with operations defined as follows (known as <math>\mathbb{F}_2</math> or <math>\Z/2</math>) is a field:

{| class="wikitable"
|-
! +
! 0
! 1
|-
! 0
| 0
| 1
|-
! 1
| 1
| 0
|}

{| class="wikitable"
|-
! *
! 0
! 1
|-
! 0
| 0
| 0
|-
! 1
| 0
| 1
|}

 

More generally, for every prime number <math>P</math>, <math>\mathbb{F}_p = \{0, 1, 2, 3, \cdots, p - 1\}</math> is a field, with operations defined by
<math>(a, b) \rightarrow a + b \mod P</math>.

An example: <math>\mathbb{F}_7 = \{0, 1, 2, 3, 4, 5, 6\}</math>, the operations are like remainders when divided by 7, or "like remainders mod 7". For example, <math>4 + 6 = 4 + 6 \mod 7</math> and <math>3 * 5 = 3 * 5 \mod 7</math>.

====Basic Properties of Fields====

'''Theorem''':
Let <math>\mathbb{F}</math> be a field, and let <math>a, b, c</math> denote elements of <math>\mathbb{F}</math>. Then:
# <math>a + b = c + b \Rightarrow a = c</math> (cancellation law)
# <math>b \ne 0 ~\&~ a * b = c * b \Rightarrow a = c</math>
Proof of 1:
1. By F4, <math>\exists b' \in \mathbb{F}, b + b' = 0</math>.
We know that <math>a + b = c + b</math>;
Therefore <math>(a + b) + b' = (c + b) + b'</math>.
2. By F2, <math>a + (b + b') = c + (b + b')</math>,
so by the choice of <math>b'</math>, we know that <math>a + 0 = c + 0</math>.
3. Therefore, by F3, <math>a = c</math>.
＾_＾

Proof of 2: more or less the same.

3. If <math>0' \in \mathbb{F}</math> is "like 0", then it is 0:
If <math>0' \in \mathbb{F}</math> satisfies <math>\forall a \in \mathbb{F}, a + 0' = a</math>, then 0' = 0.

4. If <math>1' \in \mathbb{F}</math> is "like 1", then it is 1:
If <math>1' \in \mathbb{F}</math> satisfies that <math>\forall a \in \mathbb{F}, a * 1' = a</math>, then 1' = 1.

Proof of 3 :
1. By F3 , 0' = 0' + 0.
2. By F1 , 0' + 0 = 0 + 0'.
3. By assumption on 0', 0' = 0 + 0' = 0.
＾_＾

5. <math>\forall a, b, b' \in \mathbb{F}, a + b = 0 ~\&~ a + b' = 0 \Rightarrow b = b'</math>:
In any field "<math>-a</math>" makes sense because it is unique -- it has an unambiguous meaning.
<math>(-a):</math> the <math>b</math> such that <math>a + b = 0</math>.

6. <math>\forall a, b, b' \in \mathbb{F}, a \ne 0 ~\&~ a * b = 1 = a * b' \Rightarrow b = b'</math>:
In any field, if <math>a \ne 0</math>, "<math>a^{-1}</math>" makes sense.

Proof of 5 :
1. <math>a + b = 0 = a + b'</math>.
2. By F1, <math>b + a = b'+ a</math>.
3. By cancellation, <math>b = b'</math>.
＾_＾

7. <math>-(-a) = a</math> and when <math>a \ne 0</math>, <math>(a^{-1})^{-1} = a</math>.

Proof of 7 :
1. By definition, <math>a + (-a) = 0</math>. (*)
2. By definition, <math>(-a) + (-(-a) = 0</math>.
3. By (*) and F1, <math>(-a) + a = 0</math>.
4. By property 5, <math>-(-a) = a</math>.
＾_＾

===Scanned Lecture Notes by [[User:AM|AM]]===

<gallery>
File:MAT 240 (1 of 2) Sept 10, 2014.pdf‎|page 1
File:MAT 240 (2 of 2) Sept 10, 2014.pdf‎|page 2
</gallery>