2.1. FIRST RESULTS
89
but by the associative law, the first two and the last two are equal. Thus
there are at most three different product of four elements:
a.bcd/;
.ab/.cd/;
.abc/d:
Using the associative law, we see that all three are equal:
a.bcd/
D
a.b.cd//
D
.ab/.cd/
D
..ab/c/d
D
.abc/d:
Thus there is a welldefined product of four elements, which is independent
of the way the elements are grouped for multiplication.
There are 14 ways to group five elements for multiplication; we won’t
bother to list them. Because there is a welldefined product of four or less
elements, independent of the way the elements are grouped for multiplica
tion, there are at most four distinct products of five elements:
a.bcde/;
.ab/.cde/
.abc/.de/;
.abcd/e:
Using the associative law, we can show that all four products are equal,
a.bcde/
D
a.b.cde//
D
.ab/.cde/;
etc. Thus the product of five elements at a time is welldefined, and inde
pendent of the way that the elements are grouped for multiplication.
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 Fall '08
 EVERAGE
 Algebra, Addition, Order Of Operations, Commutativity, Associative Law

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