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There is a show on our cable network called "the challenge" on which the following question was asked of high school students.

Is the set {-1, 0, 1} closed under: 1) addition 2) multiplication 3) both 4) neither

The student's answer was "both" but the moderator said the "correct" answer was "addition". I may be missing something obvious but wasn't the student correct?

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Neither the student nor the moderator gave the correct answer. – Andrés E. Caicedo Jan 26 '12 at 21:21
If these are integers with standard addition and multiplication, then this set is only closed under multiplication. Note that $1+1=2$ is not in the set. – Bill Cook Jan 26 '12 at 21:23
This set is only closed under multiplication. – Joel Cohen Jan 26 '12 at 21:23
Was the moderator a mathematician? If not, can he honestly claim to be qualified for the job? – Michael Hardy Jan 26 '12 at 21:26

Well, $1+1=2$ ... But you may think the set $\{ -1 , 0 ,1\}$ as $\mathbb{Z}/3\mathbb{Z}$, where $1+1=2=-1$. It depends on what you mean by "$+$". If you don't specify the definition of the laws, the answer is "maybe".

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For $\times$ defined "as usual", the set is stable. – Student Jan 26 '12 at 21:26

If mod-3 addition was intended, then "both" would be the right answer, but I see no reasonable way to say that it's not closed under multiplication.

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