206
Digital Electronics
The dual of AAB + AABB = A + BBBBA + BB. Therefore
AAB + AAB = A + BBBBA + BB
6.4.3 Expanded Forms of Boolean Expressions
Expanded sum-of-products and product-of-sums forms of Boolean expressions are useful not only
in analysing these expressions but also in the application of minimization techniques such as the
Quine–McCluskey tabular method and the Karnaugh mapping method for simplifying given Boolean
expressions. The expanded form, sum-of-products or product-of-sums, is obtained by including all
possible combinations of missing variables.
As an illustration, consider the following sum-of-products expression:
AAB + BBC + AABBC + AAC
It is a three-variable expression. Expanded versions of different minterms can be written as follows:
• AAB = AABBBC + CC = AABBC + AABBCC
• BBC = BBCCCA + AA = BBCCA + BBCCAA
• AABBC is a complete term and has no missing variable.
• AAC = AACCCB + BB = AACCB + AACCB.
The expanded sum-of-products expression is therefore given by
AABBC + AABBC + AABBC + AABBC + AABBC + AABBC + AABBC = AABBC + AABBC
+ AABBC + AABBC + AABBC + AABBC
As another illustration, consider the product-of-sums expression
A + BBBBA + B + C + DD
It is four-variable expression with A, B, C and D being the four variables. A + B in this case expands
to A + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DD.
The expanded product-of-sums expression is therefore given by
A + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DD
= A + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DD
6.4.4 Canonical Form of Boolean Expressions
An expanded form of Boolean expression, where each term contains all Boolean variables in their true
or complemented form, is also known as the canonical form of the expression.
As an illustration, ffAABB CC = AABBC + AABBC + AABBC is a Boolean function of three variables
expressed in canonical form. This function after simplification reduces to AAB + AABBC and loses its
canonical form.
Digital Electronics
The dual of AAB + AABB = A + BBBBA + BB. Therefore
AAB + AAB = A + BBBBA + BB
6.4.3 Expanded Forms of Boolean Expressions
Expanded sum-of-products and product-of-sums forms of Boolean expressions are useful not only
in analysing these expressions but also in the application of minimization techniques such as the
Quine–McCluskey tabular method and the Karnaugh mapping method for simplifying given Boolean
expressions. The expanded form, sum-of-products or product-of-sums, is obtained by including all
possible combinations of missing variables.
As an illustration, consider the following sum-of-products expression:
AAB + BBC + AABBC + AAC
It is a three-variable expression. Expanded versions of different minterms can be written as follows:
• AAB = AABBBC + CC = AABBC + AABBCC
• BBC = BBCCCA + AA = BBCCA + BBCCAA
• AABBC is a complete term and has no missing variable.
• AAC = AACCCB + BB = AACCB + AACCB.
The expanded sum-of-products expression is therefore given by
AABBC + AABBC + AABBC + AABBC + AABBC + AABBC + AABBC = AABBC + AABBC
+ AABBC + AABBC + AABBC + AABBC
As another illustration, consider the product-of-sums expression
A + BBBBA + B + C + DD
It is four-variable expression with A, B, C and D being the four variables. A + B in this case expands
to A + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DD.
The expanded product-of-sums expression is therefore given by
A + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DD
= A + B + C + DDDDA + B + C + DDDDA + B + C + DDDDA + B + C + DD
6.4.4 Canonical Form of Boolean Expressions
An expanded form of Boolean expression, where each term contains all Boolean variables in their true
or complemented form, is also known as the canonical form of the expression.
As an illustration, ffAABB CC = AABBC + AABBC + AABBC is a Boolean function of three variables
expressed in canonical form. This function after simplification reduces to AAB + AABBC and loses its
canonical form.