Derivation of Canonical Forms (1)
Derive canonical POS or SOP using switching algebra.
Theorem 10. Shannon's expansion theorem
(a). f(x1, x2, …, xn) = x1 f(1, x2, …, xn) + (x1)' f(0, x2, …, xn)
(b). f(x1, x2, …, xn) = [x1 + f(0, x2, …, xn)] [(x1)' + f(1, x2, …, xn)]
Example: f(A,B,C) = AB + AC' + A'C
- f(A,B,C) = AB + AC' + A'C = A f(1,B,C) + A' f(0,B,C)
= A(1×B + 1×C' + 1'×C) + A'(0×B + 0×C' + 0'×C) = A(B + C') + A'C
- f(A,B,C) = A(B + C') + A'C = B[A(1+C') + A'C] + B'[A(0 + C') + A'C]
= B[A + A'C] + B'[AC' + A'C] = AB + A'BC + AB'C' + A'B'C
- f(A,B,C) = AB + A'BC + AB'C' + A'B'C
= C[AB + A'B×1 + AB'×1' + A'B'×1] + C'[AB + A'B×0 + AB'×0' + A'B'×0]
= ABC + A'BC + A'B'C + ABC' + AB'C'