Something I noticed:
- auf / zu ---> (open / close) related to phisical mechanisms
- an / aus ---> (on / off) related to electrical devices
- "Das Fenster ist auf" (The window is open) / "Die Tür ist zu" (The door is closed)
- "Das Radio ist an" (The radio is on) / "Der Fernseher ist aus" (The television is off)
In my experience, I remember a lot of confusion betwen "auf" and "aus". It seems to me that "close" and "off" should sound similar... but it is exactly the opposite: open = auf , off = aus.
The good news are that "on" and "an" are very similar. This can be helpful. :)
'aus' means "out" as well as "off". i like to think of it as: "the television is out" as in "out of order". just a way remember it by.
'auf' has many uses and also means "by","of" and "upon".
i don't think there is a causal relation between 'aus' and 'auf'. nor a common etymology. the similarity is probably accidental.
the electricity thing is just another nice way to memorize it, nothing more.