Sum-of-Products Equations and Logic Circuits:
There are four possible technique to AND two input signals that are in the complemented and un-complemented form ( A¯ B¯ , A¯ B, A B¯ , AB) also called as fundamental products.
Below table lists each fundamental product producing high outputs.
Table: Fundamental Products for Two Inputs
|
A
|
B
|
Fundamental
Product
|
Minterms
|
|
0
|
0
|
A B
|
m0
|
|
0
|
1
|
A B
|
m1
|
|
1
|
0
|
A B
|
m2
|
|
1
|
1
|
A B
|
m3
|
For example, A¯ B¯ is high when A are B are low. The fundamental products A¯ B¯ , A¯ B, A B¯ and AB are also represented by minterms m0, m1, m2 and m3, where the suffix i of mi comes from the decimal equivalent of the binary number. There lies the advantage of understanding binary numbers before we learn Sum of Products (SOP) methods. For 3 inputs A, B and C, there are 23 minterms, m0, m1, m2, m3, m4, m5, m6 and m7 as listed in Table.
Table: Fundamental Products for Three Inputs
|
A
|
B
|
C
|
Fundamental
Products
|
Minterms
|
|
0
|
0
|
0
|
A¯ B¯ C¯
|
m0
|
|
0
|
0
|
1
|
A¯ B¯ C
|
m1
|
|
0
|
1
|
0
|
A¯ B C¯
|
m2
|
|
0
|
1
|
1
|
A¯ B C
|
m3
|
|
1
|
0
|
0
|
A B¯ C¯
|
m4
|
|
1
|
0
|
1
|
A B¯ C
|
m5
|
|
1
|
1
|
0
|
A¯ B¯ C
|
m6
|
|
1
|
1
|
1
|
A¯ B¯ C¯
|
m7
|
For instance, when A = 1, B = 1, C = 0, the fundamental product results a high output for the case Y = A B C¯ = 110¯ = 1.