Question

construct a DFA accepting the set of all strings ending in 00 over the alphabet{0,1}

Answer 1

First, you'll need to figure out how many states you'll need and give them a good and understandable name. I think your example can be done in three states. As for the name, it's best to choose something like "I've seen an 0 followed by another 0".

Once you've got that figured out, you'll need to check what happens when the input is a 1 and when the input is a 0. Which state should you go to, or do you stay in the same state?