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 thaa)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

Answer: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.