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jstarks4444
 Post subject: Smallest Element proof
PostPosted: Thu, 12 Jan 2012 04:26:43 UTC 
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Hey all, need some help with the following proof:

Let A be a non-empty subset of Z and b ∈ Z, such that for each a ∈ A, b <= a. Then A has a smallest element.

All help appreciated!
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outermeasure
 Post subject: Re: Smallest Element proof
PostPosted: Thu, 12 Jan 2012 04:39:46 UTC 
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jstarks4444 wrote:
Hey all, need some help with the following proof:

Let A be a non-empty subset of Z and b ∈ Z, such that for each a ∈ A, b <= a. Then A has a smallest element.

All help appreciated!


Use the corresponding result of \mathbb{N}.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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Shadow
 Post subject: Re: Smallest Element proof
PostPosted: Thu, 12 Jan 2012 05:37:11 UTC 
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An added hint to go with outermeasure's suggestion:

Consider the set A-b+1=\{a-b+1|a\in A\} which is a subset of \mathbb{N}, and you know c\le d\iff c-b+1\le d-b+1.

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jstarks4444
 Post subject: Re: Smallest Element proof
PostPosted: Sun, 15 Jan 2012 17:51:48 UTC 
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is there a way to use the Well-Ordering Principle and "construction" here?
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Shadow
 Post subject: Re: Smallest Element proof
PostPosted: Sun, 15 Jan 2012 19:57:56 UTC 
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jstarks4444 wrote:
is there a way to use the Well-Ordering Principle and "construction" here?


What do you mean? Both proofs we talked about make use of the well-ordering on \mathbb{N}...And what exactly is "construction"

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outermeasure
 Post subject: Re: Smallest Element proof
PostPosted: Sun, 15 Jan 2012 23:49:40 UTC 
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See also http://www.mathhelpforum.com/math-help/f7/smallest-element-proof-195165.html

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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jstarks4444
 Post subject: Re: Smallest Element proof
PostPosted: Mon, 16 Jan 2012 05:36:17 UTC 
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Why can't 'b' be negative?
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outermeasure
 Post subject: Re: Smallest Element proof
PostPosted: Mon, 16 Jan 2012 07:31:41 UTC 
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jstarks4444 wrote:
Why can't 'b' be negative?


b can be negative. Where did we state it isn't?

That's why you reduce it to \mathbb{N} by translating by -b+1 (or -b, depending on your definition of \mathbb{N}), and not just (incorrectly!) assume everything you have is nonnegative.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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