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 ![](Image385.gif)
![](Image384.gif)
has a left inverse.
- Show
is onto ![](Image385.gif)
has a right inverse.
- Show
is bijective ![](Image385.gif)
such that
and ![](Image392.gif)
.
- Let
and
be finite sets with
and let ![](Image396.gif)
be a function. Show the following conditions are equivalent.
is bijective.
is one-to-one.
![](Image397.gif)
is onto.
- Let
![](Image398.gif)
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 ![](Image404.gif)
.
- Let
and
be nonzero integers whose greatest common divisor is the integer
. Show that there are integers
and ![](Image409.gif)
such that
![](Image410.gif)