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 such that x + y = z.
(e) for every rational number z and every irrational number x, there exists a unique irrational number y such that x + y = z.
(f ) for every positive irrational number x, there is a positive irrational number y such that y < ½ and y < x.
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