4. (a) (4 pts) Explain what it means that multiplication
on the integers is associative. Give
an example of this. (You are not asked to prove that multiplication is
to explain what the term means .)
It means that (xy)z = x(yz) for any three integers x, y,
z. For example,
(b) (4 pts) Explain what it means that multiplication on
the rational numbers is associative.
Give an example of this. (You are not asked to prove that multiplication is
only to explain what the term means.)
It means that (xy)z = x(yz) for any three rational numbers
x, y, z. For example,
(c) (6 pts) Assume that multiplication is associative on
the integers. Use this to justify that
multiplication is also associative on the rational numbers. (You are now asked
that multiplication is associative on the rationals.)
Let x, y, z ∈ Q. Then there exist integers m, n, p, q, r,
s such that n, q, s ≠ 0 and x =
m/n, y = p/q, and z = r/s.
Since multiplication is associative on the integers (mp)r
= m(pr) and (nq)s = n(qs).
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