What is Boolean Algebra?
Before knowing about the Boolean theorems let us know the Boolean algebra in computer science. Boolean algebra is another name for switching algebra. Boolean algebra in computer science is employed in the analysis of digital gates and circuits. A mathematical operation on binary numbers, i.e. '0' and '1', is logical.
Basic operators in Boolean Algebra theorems include AND, OR, and NOT, among others. '.' for AND and '+' for OR are used to express operations. Operations can be done on variables represented by capital letters, such as 'A,' 'B,' and so on.
What are Boolean Theorems?
Theorems that modify the shape of a Boolean expression are known as Boolean theorems. These theorems are sometimes used to reduce the terms of the expression, and sometimes they are only used to shift the expression from one form to another.
The different types of Boolean theorems are as follows:
- De Morgan's Theorem
- Transposition Theorem
- Redundancy Theorem
- Duality Theorem
- Complementary Theorem
De Morgan's Theorem
One of the most important Boolean theorems is Demorgan’s Theorem which encapsulates two of the most significant Boolean algebra theorem rules. Suppose there are two Boolean variables A and B for these variables Demorgan’s theorem is as follows:
(A . B)' = A' + B'
As a result, the complement of variables product equals the total of its separate complements.
(A + B)' = A' . B'
Thus, the total of variables' complements is equal to the product of their individual complements.
Suppose there are three variables X, Y, and Z. According to the transposition theorem,
XY + X'Z = (X + Z) (X' + Y)
The proof of the transposition theorem is as follows:
= (X + Z) (X' + Y)
= XX' + X'Z + XY + ZY
= 0 + X'Z + XY + YZ
= X'Z + XY + YZ(X + X')
= XY + XYZ + X'Z + X'YZ
= XY + X'Z
Another one of the Boolean theorems is the redundancy theorem which is used to remove redundant terms. When a variable is related to one variable and its complement with another variable, and the following term is created by the remaining variables, the term becomes redundant.
Suppose there are three variables A, B, and C. The expression for the redundant theorem is as follows:
O = AB + BC + A’C
According to the Duality Principle, "the Dual of the expression may be accomplished by replacing the AND operator with the OR operator, as well as replacing the binary variables, such as replacing 1 with 0 and 0 with 1."
The Duality principle, also known as "De Morgan Duality," argues that "interchanging Duals pairs in Boolean algebra results in the same output of the equation." For example, the Dual of A(B+C) is as follows:
In order to get complement expression using the complementary theorem, we will have to follow certain rules that are as follows:
- Replace each OR sign with an AND sign and vice versa.
- Complement any 0s or 1s in the expression.
- Individual literals should be complemented.
For example, the Complement expression using the complementary theorem of A(B+C) is as follows:
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