Let L1 consist of strings that have an even number of 1s and 
an odd number of 0s.<p>

There is a DFA that recongizes L1.  I'll describe it below.
You may want to draw it graphically.<p>

There will be 4 states.  S00 (representing an even number of 0s and 1s
so far), S01 (representing an even number of 0s and odd number of 1s
so far), S10 (representing an odd number of 0s and even number of 1s
so far) and state S11 (representing an odd number of 0s and an odd
number of 1s so far).<p>

The start state is S00 and the only accepting state (i.e. state with a
double circle) is S10.<p>

Here is a table showing the transitions:
<pre>
State  input   | go to state
---------------|------------
S00      0     |   S10
S00      1     |   S01
S01      0     |   S11
S01      1     |   S00
S10      0     |   S00
S10      1     |   S11
S11      0     |   S01
S11      1     |   S10
</pre>