Assuming your lease runs from the first of the month to the end of the month, your landlord is correct. You would have needed to give notice 20 days before the end of February not to have to pay March rent. If your rental period is not the calendar month, you still would need to give notice 20 days before the end of the rental period in order to avoid paying for the following month.
However, the landlord does have the obligation to try to rent the property to someone else, and, if he does rent it before the end of March, you would receive your pro-rata portion of the rent for March back.
Your landlord did give you proper notice, because he notified you in February of changes that were to occur on April 1st, so you got your notice more than 20 days before the end of March.
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