Difference between revisions of "12-240/Classnotes for Tuesday September 18"

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(Recap:)
Line 4: Line 4:
 
Thrm: In a field F:
 
Thrm: In a field F:
 
1. a+b = c+b => a=c
 
1. a+b = c+b => a=c
 +
 
2. b≠0, a∙b=c∙b => a=c
 
2. b≠0, a∙b=c∙b => a=c
 +
 
3. 0 is unique.
 
3. 0 is unique.
 +
 
4. 1 is unique.
 
4. 1 is unique.
 +
 
5. -a is unique.
 
5. -a is unique.
 +
 
6. a^-1 is unique (a≠0)
 
6. a^-1 is unique (a≠0)
 +
 
7. -(-a)=a  
 
7. -(-a)=a  
 +
 
8. (a^-1)^-1 =a
 
8. (a^-1)^-1 =a
 +
 
9. a∙0=0 **Surprisingly difficult, required distributivity.  
 
9. a∙0=0 **Surprisingly difficult, required distributivity.  
 +
 
10. ∄ 0^-1, aka, ∄ b∈F s.t 0∙b=1
 
10. ∄ 0^-1, aka, ∄ b∈F s.t 0∙b=1
 +
 
11. (-a)∙(-b)=a∙b
 
11. (-a)∙(-b)=a∙b

Revision as of 21:57, 18 September 2012

Recap:

Thrm: In a field F: 1. a+b = c+b => a=c

2. b≠0, a∙b=c∙b => a=c

3. 0 is unique.

4. 1 is unique.

5. -a is unique.

6. a^-1 is unique (a≠0)

7. -(-a)=a

8. (a^-1)^-1 =a

9. a∙0=0 **Surprisingly difficult, required distributivity.

10. ∄ 0^-1, aka, ∄ b∈F s.t 0∙b=1

11. (-a)∙(-b)=a∙b