Boolean Algebra and Simplification Techniques
207
6.4.5 and Nomenclature
and notations are respectively used to represent sum-of-products and product-of-sums Boolean
expressions. We will illustrate these notations with the help of examples. Let us consider the following
Boolean function:
ffAA BB CC DD = AABBC + AABBCCD + AABBCCD + AABBCCD
We will represent this function using notation. The first step is to write the expanded sum-of-products
given by
ffAA BB CC DD = AABBCCCD + DD + AABBCCD + AABBCCD + AABBCCD
= AABBCCD + AABBCCD + AABBCCD + AABBCCD + AABBCCD
Different terms are then arranged in ascending order of the binary numbers represented by various
terms, with true variables representing a ‘1’ and a complemented variable representing a ‘0’. The
expression becomes
ffAA BB CC DD = AABBCCD + AABBCCD + AABBCCD + AABBCCD + AABBCCD
The different terms represent 0001, 0101, 1000, 1001 and 1111. The decimal equivalent of these terms
enclosed in the then gives the notation for the given Boolean function. That is, ffAA BB CC DD =
1 5 8 9 15.
The complement of ffA, B, C, DD, that is,f
(A, B, C, DD, can be directly determined from notation
by including the left-out entries from the list of all possible numbers for a four-variable function.
That is,
f
AA BB CC DD =
0 2 3 4 6 7 10 11 12 13 14
Let us now take the case of a product-of-sums Boolean function and its representation in
nomenclature. Let us consider the Boolean function
ffAA BB CC DD = B + C + DDDDA + B + C + DDDDA + B + C + DD
The expanded product-of-sums form is given by
A + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DD
The binary numbers represented by the different sum terms are 0011, 1011, 1100 and 0111 (true and
complemented variables here represent 0 and 1 respectively). When arranged in ascending order, these
numbers are 0011, 0111, 1011 and 1100. Therefore,
ffAA BB CC DD =
3 7 11 12 and f
AA BB CC DD =
0 1 2 4 5 6 8 9 10 13 14 15
An interesting corollary of what we have discussed above is that, if a given Boolean function
ffA,B,CC is given by ffAA BB CC =
0 1 4 7, then
ffAA BB CC =
2 3 5 6 and f
AA BB CC =
2 3 5 6 =
0 1 4 7
207
6.4.5 and Nomenclature
and notations are respectively used to represent sum-of-products and product-of-sums Boolean
expressions. We will illustrate these notations with the help of examples. Let us consider the following
Boolean function:
ffAA BB CC DD = AABBC + AABBCCD + AABBCCD + AABBCCD
We will represent this function using notation. The first step is to write the expanded sum-of-products
given by
ffAA BB CC DD = AABBCCCD + DD + AABBCCD + AABBCCD + AABBCCD
= AABBCCD + AABBCCD + AABBCCD + AABBCCD + AABBCCD
Different terms are then arranged in ascending order of the binary numbers represented by various
terms, with true variables representing a ‘1’ and a complemented variable representing a ‘0’. The
expression becomes
ffAA BB CC DD = AABBCCD + AABBCCD + AABBCCD + AABBCCD + AABBCCD
The different terms represent 0001, 0101, 1000, 1001 and 1111. The decimal equivalent of these terms
enclosed in the then gives the notation for the given Boolean function. That is, ffAA BB CC DD =
1 5 8 9 15.
The complement of ffA, B, C, DD, that is,f
(A, B, C, DD, can be directly determined from notation
by including the left-out entries from the list of all possible numbers for a four-variable function.
That is,
f
AA BB CC DD =
0 2 3 4 6 7 10 11 12 13 14
Let us now take the case of a product-of-sums Boolean function and its representation in
nomenclature. Let us consider the Boolean function
ffAA BB CC DD = B + C + DDDDA + B + C + DDDDA + B + C + DD
The expanded product-of-sums form is given by
A + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DD
The binary numbers represented by the different sum terms are 0011, 1011, 1100 and 0111 (true and
complemented variables here represent 0 and 1 respectively). When arranged in ascending order, these
numbers are 0011, 0111, 1011 and 1100. Therefore,
ffAA BB CC DD =
3 7 11 12 and f
AA BB CC DD =
0 1 2 4 5 6 8 9 10 13 14 15
An interesting corollary of what we have discussed above is that, if a given Boolean function
ffA,B,CC is given by ffAA BB CC =
0 1 4 7, then
ffAA BB CC =
2 3 5 6 and f
AA BB CC =
2 3 5 6 =
0 1 4 7