Boolean functions: sum of products
Example: majority function
Inputs: 3 Output: 1 whenever more than half of the inputs are true.
a b c z Minterms
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1 \abc
1 0 0 0
1 0 1 1 a\bc
1 1 0 1 ab\c
1 1 1 1 abc
z = \abc + a\bc + ab\c + abc
This can be simplified:
z = \abc + a\bc + ab (\c + c)
  = \abc + a\bc + ab
  = \abc + a\bc + ab + abc Why?
  = bc(a + \a) + a\bc + ab
  = bc + a\bc + ab + abc
  = bc + ac(\b + b) + ab
  = bc + ac + ab