- Switching Algebra Definition: Switching algebra is defined as a mathematical system for binary operations using only 1 and 0.
- Basic Operations: The primary operations in switching algebra are AND, OR, and NOT.
- Basic Laws: Fundamental laws in switching algebra include identities like A . 0 = 0 and A + 1 = 1.
- Truth Tables: Truth tables help prove the properties and laws of switching algebra.
- Simplification Methods: De Morgan’s theorem and other methods simplify complex Boolean expressions.
Boolean algebra, or switching algebra, is a mathematical logic system for performing operations in the binary system using only 1 and 0. The three basic binary operations are AND, OR, and NOT, which are used for both simple and complex operations. Boolean algebra has many rules for carrying out these operations.
In Boolean algebra, variables are represented by capital letters like A, B, and C. Each variable can only have a value of 1 or 0.
Some basic logical Boolean operations-
AND operation,
OR operation,
Not operation,
Some basic laws for Boolean Algebra,
A . 0 = 0 where A can be either 0 or 1.
A . 1 = A where A can be either 0 or 1.
A . A = A where A can be either 0 or 1.
A . Ā = 0 where A can be either 0 or 1.
A + 0 = A where A can be either 0 or 1.
A + 1 = 1 where A can be either 0 or 1.
A + Ā = 1
A + A = A
A + B = B + A where A and B can be either 0 or 1.
A . B = B . A where A and B can be either 0 or 1.
The laws of Boolean algebra are also true for more than two variables like,
Cumulative Laws for Boolean Algebra
Associative Laws for Boolean Algebra
Distributive Laws for Boolean Algebra
Redundant Literal Rule
From truth table,
| Inputs | Output | ||
| A | B | ĀB | A + ĀB |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 |
| Inputs | Output | |
| A | B | A+B |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
From truth table it is proved that,
Absorption Laws for Boolean Algebra
Proof from truth table,
| Inputs | Output | ||
| A | B | AB | A+A.B |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 |
Both the columns for A and A + A.B are the same.
Proof from truth table,
| A | B | A+B | A.X(A+B) |
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 |
Both A and A.X or A(A+B) column are same.
De Morgan’s Therem,
Proof from truth table,
Examples of Boolean Algebra
These are another method of simplifying complex Boolean expression. In this method we only use three simple steps.
- Complement entire Boolean expression.
- Change all ORs to ANDs and all ANDs to ORs.
- Now, complement each of the variable and get final expression.
By this method,
will be first complemented, i.e.
.Now, change all (+) to (.) and (.) to (+) i.e.
Now, complement each of the variable,
This is the final simplified form of Boolean expression,
And it is exactly equal to the results which have been come by applying De Morgan Theorem.
Another example,
By Second Method,
Representation of Boolean function in truth table.
Let us consider a Boolean function,
Now let us represent the function in truth table.
