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Digital Electronics
The proof of theorem 16(a) is straightforward and is given as follows:
ffXX XX YY Y Y Y Y ZZ = XXffXX XX YY Y Y Y Y ZZ + XXffXX XX YY Y Y Y Y ZZ
= XXff1 0 YY Y Y Y Y ZZ + XXff0 1 YY Y Y Y Y ZZZZTheorem 15(a)
Also
ffXX XX YY Y Y Y Y ZZ = X + ffXX XX YY Y Y Y Y ZZZZX + ffXX XX YY Y Y Y Y ZZZ
= X + ff0 1 YY Y Y Y Y ZZZZX + ff1 0 YY Y Y Y Y ZZZZTheorem 15(b)
6.3.17 Theorem 17 (Involution Law)
X = X
(6.31)
Involution law says that the complement of the complement of an expression leaves the expression
unchanged. Also, the dual of the dual of an expression is the original expression. This theorem forms
the basis of finding the equivalent product-of-sums expression for a given sum-of-products expression,
and vice versa.
Example 6.5
Prove the following:
1. LLLM + N N + LLPPQ = L + PPQQQQL + M + N NN
2. AAB + C + DDDDD + E + F FFGG = DDDAAB + CC + DDGGGE + F FF
Solution
1. Let us assume that L = XX XM + N N = Y and PPQ = Z.
The LHS of the given Boolean equation then reduces to XXY + XXZ.
Applying the transposition theorem,
XXY + XXZ = X + ZZZZX + YY = L + PPQQQL + M + N N = RHS
2. Let us assume D = XX AAB + C = Y and E + F FFG = Z.
The LHS of given the Boolean equation then reduces to X + YYYYX + ZZ.
Applying the transposition theorem,
X + YYYYX + ZZ = XXZ + XXY = DDGGGE + F F + DDDAAB + CC = RHS