Proof of the question

Question

Assume that x ∈ Q and y ∈ R∖Q. For each of the following, either prove the statement is true, or provide a counterexample to prove the statement is false.

(a) ∀x, y, x + y is rational.

(b) ∀x, y, x + y is irrational.

(c) ∀x, y, x * y is rational.

(d) ∀x, y, x * y is irrational.

Answer 1

a) False. Prove by counterexample. Let x = 0 and y = sqrt(2). x + y = 0 + sqrt(2) is irrational.

b) True. Prove by contrapositive. Let x + y = z. Show that if z is rational and x is rational then y must be rational. Let x = a/b and z = c/d, a, b, c, and d integers, b, d not equal to 0 (this must be true for x and z rational). Then, y = z - x = c/d - a/b = (cb - ad)/bd is an integer divided by an integer and therefore must be rational. Thus, z cannot be rational if x is rational and y is irrational, so if x is rational and y is irrational, then z is irrational.

c) False. Let x = 1 and y = sqrt(2). Then x * y = 1 * sqrt(2) = sqrt(2) is irrational.

d) False. Let x = 0 and y = sqrt(2). Then, xy = 0(sqrt(2)) = 0 is not irrational.