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Susan123456
 Post subject: Quantifiers
PostPosted: Fri, 11 Sep 2009 01:43:21 UTC 
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Could someone show me how to prove the following?

(not for some x)P(x) is equivalent to ((for all x)(not P(x))

Thank you so much

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outermeasure
 Post subject: Re: Quantifiers
PostPosted: Fri, 11 Sep 2009 01:50:38 UTC 
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Susan123456 wrote:
Could someone show me how to prove the following?

(not for some x)P(x) is equivalent to ((for all x)(not P(x))

Thank you so much


"(not for some x)P(x)" is: \neg(\exists x) P(x)
"(for all x)(not P(x))" is: (\forall x)\neg P(x)

By definition, (\exists x) is a shorthand for \neg(\forall x)\neg, and now apply double negation...

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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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