In monopoly, if a player owns all of a set of properties (e.g. all three green properties) then the (unimproved) rent of those properties is doubled. Imagine now that one of the properties is mortgaged and somebody lands on one of the unmortgaged properties in the set. Is the rent still doubled even though one of the properties is mortgaged? On the one hand, the player still owns all three properties. On the other hand, only 2 of the properties are "active" in the sense that they aren't mortgaged.
Yes, rent is doubled even if some of the properties are mortgaged.
It is an advantage to hold all the Title Deed cards in a color-group (e.g., Boardwalk and Park Place; or Connecticut, Vermont and Oriental Avenues) because the owner may then charge double rent for unimproved properties in that color-group. This rule applies to unmortgaged properties even if another property in that color-group is mortgaged.