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I am reading a book, and I am trying to understand what the writer really mean by the following terms. I would like to understand what these words mean in relation to the examples.

In regular algebra, addition and multiplication are commutative: $$A + B = B + A$$ $$A \times B = B \times A$$ they are also associative: $$A + (B + C) = (A + B) + C$$ $$A \times (B \times C) = (A\times B) \times C$$ And multiplication is said to be distributive: $$A \times (B + C) = (A \times B) + (A \times C)$$


In Boolean algebra, the $+$ operator is distributive over the $\times$ operator: $$W + (B \times F) = (W + B)\times (W + F)$$ $$W = \text{white}\qquad B = \text{black}\qquad F = \text{female}$$

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    What part of this don't you understand?2011-12-28
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    There is a typo, when you say multiplication is said to be distributive, in symbols that should be $A\times (B+C)=A\times B+A\times C$.2011-12-28
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    Could you be more precise about your question?2011-12-28
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    @AndréNicolas: Thanks. I have updated the question.2011-12-28
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    I would like to understand what are these words mean in relation to the examples he gives.2011-12-28
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    The meaning of the words are the formulas that come right after the colon in each case -- no more, no less.2011-12-28
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    @HananN. Are you asking how the mathematical concepts relate to the meaning of the words in the English language?2011-12-28
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    Commutative: the result is the same if operands are *commutated*. $x + y$ is the same as $y + x.$ Assosicative: $x + y + z$ is the same as $(x + y) + z$ as $x + (y + z)$. It doesn't matter if you *assigned an order* (also: *associate*) to $(x+y)$ before $(y + z)$ and vice versa. Distributive: results are the same when you *distribute*.2011-12-29
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    @J.D. : Just add some explenatiin about the Boolian at the end of the question, and you get the crown here. Thanks.2011-12-29
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    @HenningMakholm: I know how to read, and the reason i have asked that question, is to get more sense, and maybe to apply the consents to a different instances. I know that the examples are actually the explanation of the words, but what are they is the question, and i think that J.D. on the comment above give me that.2011-12-29
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    @Hanan: They are not _examples_ -- they are (except for the aribitrary meaning assignments to $W$, $B$, and $F$) _definitions_. This is a crucial distinction.2011-12-29
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    (Note that there's an implicit "for all $A$, $B$, and $C$" in front of each of the defining equations).2011-12-29
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    @HananN. In layman's terms: Obviously you have a situation where an *algebra* is defined over {B, W, F}. That is, the operators $\times$ and $+$ operate on ${B, W, F}.$ In this case, the same meaning of distributive hold here. (That's why studying *algebra* is interesting: the properties of the operators hold *regardless* of the underlying semantics of the operators.)2011-12-30

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