Let I {\displaystyle \mathbb {I} } be the set of irrational numbers. That is I = R − Q {\displaystyle \mathbb {I} =\mathbb {R} -\mathbb {Q} } .
Decide whether the following statements are true or false. Prove your answers. Hint:The results from part (a) will be of great help.
(i) ∀ x ∈ I , ∀ y ∈ I , x y ∈ I {\displaystyle \forall {}x\in \mathbb {I} ,\forall {}y\in \mathbb {I} ,xy\in \mathbb {I} } .
(ii) ∀ x ∈ I , ∃ y ∈ I s.t. x y ∈ I {\displaystyle \forall {}x\in \mathbb {I} ,\exists {}y\in \mathbb {I} \quad {\textrm {s.t.}}\quad {}xy\in \mathbb {I} } .
be the set of irrational numbers. That is 
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