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Revision as of 14:30, 7 May 2009
, 14:30, 7 May 2009→Commentary Note 12.5
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The products commute, so the equation holds. In essence, the matrix identity turns on the fact that the law of exponents <math>(a^b)^c = a^{bc}\!</math> in ordinary arithmetic holds when the values <math>a, b, c\!</math> are restricted to <math>\mathbb{B}.</math>
The products commute, so the equation holds. In essence, the matrix identity turns on the fact that the law of exponents <math>(a^b)^c = a^{bc}\!</math> in ordinary arithmetic holds when the values <math>a, b, c\!</math> are restricted to the boolean domain <math>\mathbb{B}.</math> Interpreted as a logical statement, the law of exponents <math>(a^b)^c = a^{bc}\!</math> amounts to a theorem of propositional calculus that is otherwise expressed in the following ways:
{| align="center" cellspacing="6" width="90%"
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<math>\begin{matrix}
(a >\!\!\!-~ b) >\!\!\!-~ c & = & a >\!\!\!-~ bc
\\[8pt]
c ~-\!\!\!< (b ~-\!\!\!< a) & = & cb ~-\!\!\!< a
\\[8pt]
c \,\Rightarrow\, (b \,\Rightarrow\, a) & = & c \land b \,\Rightarrow\, a
\end{matrix}</math>
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==References==
==References==
