Problem

Prove that (a) if x + y is irrational, then either x or y is irrational. (b) i...

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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Solutions For Problems in Chapter 1.7