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Construct an argument that shows that the set of rational numbers is closed under division. That is, if x and y are

rational numbers (with y nonzero) and ???? = x / y' prove that ???? must also be a rational number.

1 Answer

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Answer:


(x)/(y) can also be expressed as a ratio of two integers.

Explanation:

Assume x and y are rational number, then x and y can be written as:


x=(a)/(b), where a and b are integers and b≠0


y=(c)/(d), where c and d are integers and d≠0

Then,


(x)/(y) =
((a)/(b) )/((c)/(d) )

=
(a*d)/(b*c)

Since the set of integers are closed under multiplication, a*d is also an integer. Similarly b*c is also an integer.


(x)/(y) can be expressed as a proportion of two integers, provided that y≠0. From this we can conclude that
(x)/(y) is a rational number, which concludes that the set of rational numbers are closed under division.

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