Analyse the following statements concerning real numbers and provide detailed proofs or counter-examples as required.
(i) S:∀x∈R∃y∈R(y2=x). Decide whether S is true or false and justify your answer.
(ii) If S is false, exhibit one explicit real number that serves as a counter-example and explain why it invalidates the statement.
(iii) Modify the domain of x so that the sentence ∀x∃y(y2=x) becomes true. State the revised domain and prove the truth of the resulting statement.
(iv) Consider the reordered quantification T:∃y∈R∀x∈R(y2=x). Determine the truth value of T and justify your conclusion.