Math 420/620

Problem Set 1

 

  1. Prove Proposition 0.1 from the notes. That is, let
be a function.
  1. Show that is one to one
has a left inverse.
  1. Show is onto has a right inverse.
  2. Show is bijective such that and
.

 

  1. Let and be finite sets with and let
be a function. Show the following conditions are equivalent.
  1. is bijective.
  2. is one-to-one.
is onto.

 

  1. Let
be a nonempty set.
  1. If is an equivalence relation on , then the set of equivalence classes of form a partition of .
  2. If is a partition of then there is an equivalence relation on whose equivalence classes are precisely the sets
.

 

  1. Let and be nonzero integers whose greatest common divisor is the integer . Show that there are integers and
such that