Does an 8th note receive half a beat

The "time signature" at the beginning of the piece tells how many beats
there will be in each measure, and what size note will receive one beat. 
From there on, all the different size notes in the piece are figured from that. 
It can even change during the piece, but if that happens, then there has to be
a new time signature at the place in the music where it changes.

May consider a group  g=({0,1},+)g=({0,1},+)where  ++  will work as the xor operator like we work in boolean algebra. note here, that ‘00’ and ‘11’ are not the usual 0 and 1 that we work with everyday. the elements are  called  ‘00’ and ‘11’ and  may not mean  zero (nothing) and one (single). ‘00’ is the identity for the given binary operator +, meaning that any element operated with ‘00’ gets us that element itself.  for a set  ss  in  g′g′  having more elements than just  {0,1}{0,1}  the values for  a+b∈sa+b∈s, such that  a∈s,b∈sa∈s,b∈s, have to be first defined, only then can you determine the value of  1+11+1. the answer could possibly have been  22  if oanswer is25,000

Probably north korea if your family has decided to use only nuclear power.