Consider a set of strings {a b c} such that a,b,c belongs to {0,1}. There are $2^3$ such strings. Now define a set of functions from {a b c} $\rightarrow$ {0,1} such that,
- The functions are balanced ( Image should have an equal number of zeroes and ones)
- Any two functions should give the same value for exactly $2^2$ strings. Out of those $2^2$ strings, $2^1$ strings should give value 0 and the other $2^1$ strings should give value 1.
If these are the conditions, then how many functions are there? The answer I ended up with is 7.
How do I prove it?
I got these 7 functions by taking the XOR combinations of the bits (The functions are a, b, c, a$\oplus$b, b$\oplus$c, a$\oplus$c, and a$\oplus$b$\oplus$c)
How do I prove these are the functions?
How do I generalize this to higher dimensions? Say for an 'n' bit string?