(Conceptual clarity) Answer the following questions on the idea of disproving a universal statement.
(i) Define the terms (a) conjecture and (b) counterexample as used in mathematical reasoning.
(ii) Explain in your own words why a single valid counterexample is sufficient to show that a universal statement of the form “For all …” is false.
(iii) The statement S is: “The square of every odd integer is even.” State whether S is true or false. If it is false, exhibit one specific integer that refutes it and explain clearly how this example invalidates the whole statement.