14-240/Tutorial-Sep30: Difference between revisions
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Find a set <math>S</math> of two elements that satisfies the following: |
Find a set <math>S</math> of two elements that satisfies the following: |
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Solution: |
Solution: |
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We verify that <math>S</math> satisfies the (1) and (2). |
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==Nikita== |
==Nikita== |
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Revision as of 21:21, 4 October 2014
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Boris
Problem
Find a set [math]\displaystyle{ S }[/math] of two elements that satisfies the following:
(1) [math]\displaystyle{ S }[/math] satisfies all the properties of the field except distributivity.
(2) [math]\displaystyle{ \exists x \in S, 0x \neq 0 }[/math].
Solution:
Let [math]\displaystyle{ S = \{ a, b \} }[/math] where [math]\displaystyle{ a }[/math] is the additive identity and [math]\displaystyle{ b }[/math] is the multiplicative identity and [math]\displaystyle{ a \neq b }[/math]. After trial and error, we have the following addition and multiplication tables:
| [math]\displaystyle{ + }[/math] | [math]\displaystyle{ a }[/math] | [math]\displaystyle{ b }[/math] |
|---|---|---|
| [math]\displaystyle{ a }[/math] | [math]\displaystyle{ a }[/math] | [math]\displaystyle{ b }[/math] |
| [math]\displaystyle{ b }[/math] | [math]\displaystyle{ b }[/math] | [math]\displaystyle{ a }[/math] |
| [math]\displaystyle{ \times }[/math] | [math]\displaystyle{ b }[/math] | [math]\displaystyle{ a }[/math] |
|---|---|---|
| [math]\displaystyle{ b }[/math] | [math]\displaystyle{ b }[/math] | [math]\displaystyle{ a }[/math] |
| [math]\displaystyle{ a }[/math] | [math]\displaystyle{ a }[/math] | [math]\displaystyle{ b }[/math] |
We verify that [math]\displaystyle{ S }[/math] satisfies the (1) and (2).
