U.S. presidents

Discussion Problem:Let  be the set of all calendar dates from 4/30/1789 to 9/25/2014.Let  be the set of names of U.S. presidents.Let  be the set of names of first ladies of the U.S.Consider the following two functions  where  is the name of the president of the U.S. on date  (the incumbent on inauguration days, to avoid ambiguity) and  where  is the name of the first lady of the U.S. during the presidency of president  .1. Can the composition be defined? What about  ?2. Which of the following functional inverses exist and which do not? ,  ,  ,