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15 September, 09:38

For each of the following statements, determine whether the statement is true or false. For a true statement, give a proof. For a false statement, write out its negation and prove tha

a) For all rational numbers x, there is a positive integer n so that nx is an integer.

b) There is a positive integer n so that for all rational numbers x, nx is an integer

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  1. 15 September, 10:39
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    a) True

    b) False

    Step-by-step explanation:

    a) True:

    Let's set x as a rational number. We can rewrite x as p/q, with p an integer number and q a natural.

    Let's notice that q is a positive integer (that's the definition of natural), so q*x = q*p/q = p is an integer.

    q is the n we were looking for.

    b) False:

    Let's prove it by contradiction. We will assume there is a positive integer n that for all rational numbers x, n*x is an integer.

    Let's take x = 1/n+1. By hypothesis, n*x = n*1/n+1 = n/n+1 has to be an integer.

    The only way for that to be true is if n+1 = 1, but that means n = 0, that's absurd because n was a positive number.

    That's why the statement is false.
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