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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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You've quoted the definitions right there; there's not much more to understanding beyond just getting a lot of experience with manipulating expressions using those rules to gain some intuition of their utility.

For instance, you may figure out that if you have an operation * which is commutative but not associative, then the following manipulations are valid:

$a * (b * c) = a*(c*b) = (b*c) * a = (c*b)*a$

but the following is not:

$a*(b*c) = b*(a*c)$

The effect of associativity is to allow you to drop the parentheses altogether without worrying about ambiguity of the expression. So there are 5 different interpretations of the expression $a*b*c*d$, namely, $(a*b)*(c*d), ((a*b)*c)*d, (a*(b*c))*d, a*((b*c)*d)$ and $a*(b*(c*d))$. Associativity guarantees that these are all the same.

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Let us examine the associative property first.

  1. Associative property of addition is how numbers are associated or
    grouped together. We need a convention on how to compute 1+(2+3). The
    associative property shows that 1+(2+3) yields same value as (1+2)+3.

    For instance,

          ( *   **)   ***     *   (**    ***)
          (1 +  2 ) +  3   =  1 + (2   +  3 ) 
          ( A + B ) +  C   =  A + (B   +  C ) 
    
  2. Consider the following diagram where we have three 4s or four 3s which depends on if we group by rows or by columns.

     * * * *
     * * * *
     * * * *
    

    Thus commutative principle for multiplication shows that the order does not matter and 3 X 4 = 4 x 3 or the general form:

    (A)(B)$\equiv$(B)(A)

  3. For distributive principle of multiplication consider the following array of counters to represent (3)(12):

     * * * * * * * * * *  | * *
     * * * * * * * * * *  | * *
     * * * * * * * * * *  | * *
    

    The rectangular array is divided into two rectangles such as the
    first one is (3)(10) and the next one is (3)(2). The multiplication has been broken into two parts ie distributed. The product (3)(10+2) can be replaced as (3)(10) + (3)(2).

    The full statement is:

    Multiplication is distributive with respect to addition.

    Symbolically, it can written as: (A) (B+C) $\equiv$ (A) (B) + (A) (C)

    Multiplication is also distributive with respect to subtraction.

    Thus, (A) (B-C) $\equiv$ (A) (B) - (A) (C)

Reference:

Lay, L.Clark. The Study of Arithmetic. The Macmillan Company. 1966.

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Commutatitive:Is a condition where by the number's and letter's are adding or multiply. Example addition; x+y=y+x or 2+3=3+2 and multiply; xy=yx or 2*3=3*2

Associative:Is a condition that a group of quantities connecte by operators gives the same result whatever their grouping. Example addition, a+(b+c)=(a+b)+c or multiplication, a*(bc)=(ab)*c.

Distributive:Is a matter of separate or break it into parts,and it makes the number to a group or letter to a group to multiply and add easly. Example a+(bc)=(a+b)(a+c) or 2+(3*4)=(2+3)*(2+4).