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

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.