Let H xpx elementof Q and p is prime Give a CONCISE descrip

Let H = {x/p|x elementof Q and p is prime}. Give a CONCISE description of H. Prove that your description in (a) is correct.

Solution

1

a)

H is the set of all rational numbers

a)

Consider any rational number, y

y is of the form :p/q where, p,q are integer

Now any integer, q has a prime factor,

Let, z be it\'s one prime factor

So, q=mz

So we can write, x=p/(mz)

Hence,x is in H

So all rational numbers are in H

Now consider any element in H

It is a ratio of a rational number and a prime number hence any element in H is a rational number.

Hence, H=Q