need help with the proofs of these two Please be very detail

need help with the proofs of these two. Please be very detailed so i can understand. thank you

EXERCISES J6. 6. Prove For alla, b R.ifac Qanda b is irrational, then b is irational. Prove. For aab.R.if. . Prove For all a bR ifaQa 0, and b is imational, then ab is

Solution

6. Given that a is rational and a+b is irrational. We have to show that b is irrational.

Suppose b is rational, then sum of two rational numbers is again a rational number.

This means that a+b is a rational number. Which is a contradictation to the fact that a+b is irrational.

Hence, b must be irrational.

7.Given that a a non zero rational number and b is irrational. We have to show that ab is irrational.

Suppose that ab is rational, then ab/b=b is also rational number.Which is a contradictation to the fact that b is irrational.

Hence,a b must be irrational.