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January 23rd, 2013, 07:52 PM   #1
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What is \mathbb{R}-\{-1\}?

I have been asked to help a friend with his homework. He has questions, but no textbook, so I can't look up what symbols mean. The question is

Show that is a group under the operation

What is ? Is it all reals except for negative one? If yes, how on earth do I prove that it's a group?

Thanks.
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January 23rd, 2013, 08:25 PM   #2
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Re: What is \mathbb{R}-\{-1\}?

That is a standard interpretation of the symbols.
You prove something is a group by...
Wait for it...
The DEFINITION!

Closed under operation "*"
Has identity
Has inverse
Associative
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January 23rd, 2013, 09:11 PM   #3
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Re: What is \mathbb{R}-\{-1\}?

Quote:
Originally Posted by The Chaz
...You prove something is a group by...
Wait for it...
The DEFINITION!

...
I knew THAT. I wasn't sure how to do those steps. But it was easy once you confirmed the definition.
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January 23rd, 2013, 09:13 PM   #4
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Re: What is \mathbb{R}-\{-1\}?

So what's the inverse of element "c"?
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January 23rd, 2013, 09:29 PM   #5
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Re: What is \mathbb{R}-\{-1\}?

Quote:
Originally Posted by The Chaz
So what's the inverse of element "c"?
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January 23rd, 2013, 10:53 PM   #6
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Re: What is \mathbb{R}-\{-1\}?

in fact, ?:R-{0}--->R-{1} given by ?(x) = x-1 is a group isomorphism, since:

?(xy) = xy - 1 = xy + x - x + y - y - 1 = xy - x - y + 1 + (x-1) + (y-1) = (x-1)(y-1) + (x-1) + (y-1) = (x-1)*(y-1) = ?(x)*?(y).

this gives us a way to find c^-1:

it is: ?(1/(?^-1(c))) = ?(1/(c+1)) = 1/(c+1) - 1
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