My question involves landlord-tenant law in the State of: Arizona

My year lease was going to be up on 04/30/2013, with it automatically going to a month-to-month option.

On 04/09/2013, I gave a 30 day notice to move out on 05/09/2013. I was told that because I did not give my notice on the 04/01 to move out by 04/30, that I had broken my lease. They are now holding my security deposit and telling me that I must pay the entire month of May's rent (citing ARS 33-1375).

When I explained that I did give a 30 day notice, they disagreed, stating that they do business on the calendar month, so I was obligated to give my notice on the first.

I re-read my lease, and it does not state anywhere that the notice has to be on the first, it merely states that I have to give a 30 day notice.

They pro-rated my rent when I moved in 2 days before the first last year, I don't understand why they can't pro-rate May's rent 9 days.

Am I wrong in my thinking that I gave a 30 day notice? All of their charges stem from their claim that I did not give a 30 day notice.

Thanks in advance for any advice you can give me - I plan on fighting this, but I obviously don't want to waste anyone's time if I am in error.