Also, if mode "C" is ALT, is mode "A" ON?
What about mode "S"
I believe modes "C", "A" & "S" are not available in most light aircraft, but it would be good to know what they are...
The truth is in fact a bit more complex than you're being told. Mode A, Mode C and Mode S are actually interrogation modes, sequences of bits/pulses sent by the ground radar to tell the transponder what data to put in its response.
In response to a Mode A interrogation, it returns the 4 octal digit code.
In response to a Mode C interrogation, it returns encoded altitude, provided that there is an altitude encoding input and the transponder is set to ALT. If there is no altitude input or it is set to ON, it does respond to Mode C interrogations, but with a blank "data unavailable" response. TCAS interrogations are Mode C.
There are different Mode S interrogations, some of which are all-call (all or many transponders reply), some of which are selective (only the transponder with the correct hex code replies).
A "Mode S transponder" is capable of replying to Mode A and Mode C interrogations, and even helpfully suppresses them if it is in ground mode. Older ("Mode A/C" or sometimes "ATCRBS") transponders reply to some Mode S interrogation modes.