Let P(x, y) mean that x is a reciprocal of y.

Question

a) Let Q⁺ be the set of positive rational numbers. Consider the following statement: ∀x ∈ Q⁺, ∃y ∈ Q⁺ such that P(x, y). Is this statement true or false? Explain.

b) Now change Q⁺ to Q, the set of all rational numbers. Is this statement true or false? Explain.

Answer 1

a) If Q+ is the set of all positive rational numbers then it contains also all positive integers.

Therefore, for any x in Q+ there is an y in Q+ such that P(x,y), thus, the statement is true.

For instance, if you take x=3 then the reciprocal y=1/3 is still a positive rational number (in Q+).

If you consider x=1/2 then the reciprocal y=2 is still a positive rational number (in Q+)

b) If you consider Q instead of Q+ the same argument applies to the following statement :

for all x in Q there exist y in Q such that P(x,y).

Best,

Davide