Math 420/620
Problem Set 1
Prove Proposition 0.1 from the notes. That is, let
be a function.
- Show that is one to one
has a left inverse.
- Show is onto has a right inverse.
- Show is bijective such that and
.
- Let and be finite sets with and let
be a function. Show the following conditions are equivalent.
- is bijective.
- is one-to-one.
is onto.
- Let
be a nonempty set.
- If is an equivalence relation on , then the set of equivalence classes of form a partition of .
- If is a partition of then there is an equivalence relation on whose equivalence classes are precisely the sets
.
- Let and be nonzero integers whose greatest common divisor is the integer . Show that there are integers and
such that