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:
$$W + (B \times F) = (W + B)\times (W + F)$$
$$W = \text{white}\qquad B = \text{black}\qquad F = \text{female}$$
Best Answer
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.