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PostPosted: Tue, 5 Oct 2010 15:49:00 UTC 
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How many different Boolean functions F(x,y,z) are there such that
F([x],y,z) = F(x,[y],z) = F(x,y,[z])
for all values of the Boolean variables x, y, and z?

[] means the complement of (-)

I don't even understand the question.. I would understand if it asked something like "..F(x,y,z) = xy +z" .. but sadly it does not.


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PostPosted: Tue, 5 Oct 2010 16:20:17 UTC 
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Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
mred90 wrote:
How many different Boolean functions F(x,y,z) are there such that
F([x],y,z) = F(x,[y],z) = F(x,y,[z])
for all values of the Boolean variables x, y, and z?

[] means the complement of (-)

I don't even understand the question.. I would understand if it asked something like "..F(x,y,z) = xy +z" .. but sadly it does not.


Just count.

Hint: Write F in full disjunctive normal form, and note conditions such as F(x,y,z)=1 implies F contains xyz+\bar{x}\bar{y}z+\bar{x}y\bar{z}+x\bar{y}\bar{z}. Similarly ... and hence you only have free choice for F(0,0,0) and F(1,1,1) and so there are ... such functions.

_________________
\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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 Post subject: Re: Foundations
PostPosted: Wed, 20 Oct 2010 12:39:46 UTC 
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http://www.asic-world.com/digital/kmaps2.html


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