Boolean algebra
Boolean algebra
In real world, devices such as calculators are considered as magical devices that perform complex calculations in a fraction of seconds. It is because the electronic devices in digital systems are based on Boolean algebra. Boolean algebra is significantly different from conventional algebra. In digital systems, logical operations are performed using Boolean algebra. Microcontrollers or other programmed components are used to perform logical operations in electronic devices.
Logical Operations
Different logical operations are briefly discussed below:
- AND operation
It is similar to multiplication in conventional algebra. It is represented by •, ?, ?.
- OR operation
It is similar to an addition in conventional algebra. It is represented by +, V, U.
- NOT operation
It is similar to complement or inversion. It is represented by an over-score or bar ‘-‘over the variable.
- NAND operation
It is a combination of AND plus NOT operation. Simply negation of AND operation. The word ‘NAND’ can be read as “NOT + AND.”
- NOR operation
It is a combination of OR plus NOT operation. Simply negation of OR operation. The word ‘NAND’ can be read as “NOT + OR.”
- X – OR and X-NOR operations
The X-Or and X-NOR operation on variables P & Q in Boolean algebra is denoted by P ? Q (=PQ’ +P’Q) and P ? Q (= PQ + P’Q’), respectively. The word ‘X-OR’ can be read as “Exclusive OR.” While the word ‘X-NOR’ can be read as “Exclusive NOR.”
Laws of Boolean Algebra
The Boolean algebraic laws play a very important role when a designer wants to reduce the total number of logic gates without affecting the output and also to simplify Boolean expressions. There are different types of Laws of Boolean Algebra, some popular laws are given below:
1. Commutative Law
This law allows the change of position of AND or OR operation variables. The order is immaterial according to this law. It can be applied to any ‘n’ number of variables.
A+B= B+A
A•B = B•A
2. Associative Law
This law allows the grouping of two variables. The order of grouping of variables is immaterial. It is applicable to any ‘n’ number of variables.
A+ (B+C) = (A+B) + C = A+B+C
A• (B•C) = (A•B) C = ABC
3. Distributive Law
This law allows the multiplication of expressions. It is applied to any ‘n’ number of variables.
A (B+C) = AB+BC
(A+B) (C+D) = AC+AD+BC+BD
4. Involution Law
This law allows the simplification of variables from complemented form. It is applied to any ‘n’ number of variables.
(A’)’ = A
5. Absorption Law
This law allows converting expression in simplest form by absorbing similar terms. It is applied to any ‘n’ number of variables.
A +AB = A (1+B) = A
A (A+B) = (A+0). (A+B) = A+ (0.B) = A
Boolean Algebraic Theorems
6. AND-operation theorem
A•A=A
A•0=0
A•1=A
A•A’=A
7. Involution Theorem
(A’)’ = A
8. OR- operation theorem
A + A = A
A + 0 = A
A + 1 = 1
A + A’ = 1
9. De Morgan’s theorem
Among all other theorem’s, this theorem is widely used in many applications. This theorem is very useful to simplify the expressions in which a sum or product is complemented. It applies to any ‘n’ number of variables. It is very helpful to remove long over-bars in any given logical expression.
(A•B)’ = A’+ B’
(A+B)’ = ( A’• B’)
10. Transposition theorem
(A+B)(A+C) =A + BC
11. Distribution Theorem
A+BC = (A+B)(A+C)
12. Consensus theorem
It is generally used to eliminate the redundant term. It contains three variables in which each variable is used two times, and only one variable is in uncomplemented or complemented form.
AB + A’C + BC = AB +A’C
Here BC is a redundant term.
13. Duality theorem
It is similar to the negative logic of the given relation. Simply we've got to alter each OR sign by AND sign, and it's vice-versa. Also, complement all the ‘0’ or ‘1’ appearing in the expression. And keep the variables unchanged.
A’BC + ABC’ +AB’C’ = (A’ + B + C) (A+B+C’) (A+B’+C’)
14. Complementary theorem
Simply we have to change each OR sign by AND sign, and it's vice-versa. Complement any ‘0’ or ‘1’ appearing in expression. Also, complement the individual variables.
Complement of (AB’C +A’BC’+ AB’C’) = (A’+B+C’) (A+B’+C)(A’+B+C)
Some important theorems are summarized below.
| Expression | Resultant |
| A + 0 | = A |
| A . 1 | = A |
| A + 1 | = 1 |
| A . 0 | = A |
| A + A | = A |
| A . A | = A |
| A + | = 1 |
| A . | = 0 |
| A . (B + C) | = AB + AC |
| A . BC | = (A + B) (A + C) |
| A + AB | = A |
| A ( A + B) | = A |
| A + | = (A + B) |
| A( | = AB |
| AB + A | = A |
| (A + B) . (A + | = A |
| AB + | = (A + C) ( |
| (A + B) ( | = AC + |
| AB + | = AB + |
| (A + B) ( | = (A + B) ( |
| = | |
| = |