Given two arbitrary terms , do the following terms always exist?

• A term such that and
• A term such that and

For each question, determine if such a term always exists, if it sometimes exists, or if it never exists. Provide a convincing argument for your answer:

• if it always exists, then provide a construction for that term depending on .
• if it sometimes exists, provide values for for which the term exists and for which the term does not exist.
• if it never exists, provide a proof that this is the case.

In your solution, you are allowed to use any results, facts or procedures explained in the lecture notes. Explain clearly what your answer for each question is (always, sometimes, or never), followed by your argument. Solutions are only accepted as a PDF file.