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I would like to know what the difference between "the key to the door" and "the the key of the door", or between "the servant to the master" and "the servant of the master" is.

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English seems to have a preference for to or for when referring to an item for "operating" something. Hence:

the key to/for the door

the switch to/for the hall lights

the catch to/for the window

On the other hand, 'of' is used more when you're naming a component part of something; sometimes either 'of' or 'to/for' is possible, depending on whether you're just naming a part of something or implying its intended use:

the leg of the table

the door of/to/for the fridge

the ingredients of/to/for this soup

When identifying a part of something, English also permits other prepositions denoting location:

the handle on the fridge

the largest collection in the world

Various other languages tend to use the equivalent of 'of' in many of these cases: i.e. they have less of a distinction between naming the 'utility' vs 'existence' vs 'identification/location' of a component of something.

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In English we speak of the key to the door, not the key of the door.

With respect to servants, you can perform as the servant to the master or be the servant of the master. You would use to when describing the act of performing as a servant, and of when describing the state of being a servant.

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This can be surprising for who speaks English as second language, but in English it is key to the door, while other languages would say the equivalent of key of the door. – kiamlaluno Feb 17 '11 at 18:06
Plus, one rarely hears key to the door even in regular conversation. We simply say key or key to [name of building or room]. E.g. "Where's the key?" "I need the key to the library" "Where's the key to the locker room?" Most times, we tend to form compound nouns with key, as in: "gym keys", "car key[s]", "garage key", "cellar key", etc. – Jimi Oke Feb 17 '11 at 20:06

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