Difference between revisions of "12-240/Classnotes for Tuesday September 18"
From Drorbn
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Thrm: In a field F: | Thrm: In a field F: | ||
1. a+b = c+b => a=c | 1. a+b = c+b => a=c | ||
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2. b≠0, a∙b=c∙b => a=c | 2. b≠0, a∙b=c∙b => a=c | ||
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3. 0 is unique. | 3. 0 is unique. | ||
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4. 1 is unique. | 4. 1 is unique. | ||
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5. -a is unique. | 5. -a is unique. | ||
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6. a^-1 is unique (a≠0) | 6. a^-1 is unique (a≠0) | ||
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7. -(-a)=a | 7. -(-a)=a | ||
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8. (a^-1)^-1 =a | 8. (a^-1)^-1 =a | ||
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9. a∙0=0 **Surprisingly difficult, required distributivity. | 9. a∙0=0 **Surprisingly difficult, required distributivity. | ||
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10. ∄ 0^-1, aka, ∄ b∈F s.t 0∙b=1 | 10. ∄ 0^-1, aka, ∄ b∈F s.t 0∙b=1 | ||
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11. (-a)∙(-b)=a∙b | 11. (-a)∙(-b)=a∙b |
Revision as of 21:57, 18 September 2012
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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