x and y are rational numbers w and z are irrational numbers

x and y are rational numbers, w and z are irrational numbers. Prove the following or give a counterexample to show it is a false statement: x times y is rational, w times z is irrational, w plus z is irrational

Solution

x,y are rational then x times y is rational

True

Proof

x, y are rational so x=p/q and y=r/s

So, xy=(pq)/(qs) hence rational

w,z are irrational then w times z is irrational

False

Counterexample

Let, w=z=sqrt{2}

wz=2 which is rational

w,z are irrational then w+z is irrational

False

Counterexample

LEt, w=sqrt{2},z=-w

w+z=0 which is rational