(a) Prove that if x is rational and y is irrational, then x + y is irrational.
(b) Prove that there exist irrational numbers x and y such that x + y is rational.
(c) Prove that for every rational number z, there exist irrational numbers x and y such that x + y = z.
(d) Prove that for every rational number z and every irrational number x, there exists a unique irrational number y such that x + y = z.
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