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Revision as of 02:40, 8 May 2009
, 02:40, 8 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 the boolean domain <math>\mathbb{B}.</math> Regarded 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:
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} = \{ 0, 1 \}.</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:
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