A months notice of cancellation and paying twice

I visited ADT offices on the 19/05 to tell them about my cancellation of the services as i had moved out of the place. Now they told me i have to put up a notice as explained when i signed up for services. Now they want me to pay for May and also for June as their month runs from month end to month end. I find this absurd and very unfair to me, i even dare the day i signed up with ADT. How must i pay for 2 months because of my D/O running from 20th to 20th.0