Chapter 1.7, Problem 5E

Prove that
(a) if x + y is irrational, then either x or y is irrational.
(b) if x is rational and y is irrational, then $x+y$ is irrational.
(c) there exist irrational numbers x and y such that $x+y$ is rational.
(d) for every rational number z, there exist irrational numbers x and y suchthat $x+y=z$ .
(e) for every rational number z and every irrational number x, there exists aunique irrational number y such that $x+y=z$ .
(f ) for every positive irrational number x, there is a positive irrational numbery such that $y<\frac{1}{2}$ and $y<x$ .