Boolean algebra is a type of algebra that is created by operating the binary system. In the year 1854, George Boole, an English mathematician, proposed this algebra. This is a variant of Aristotle's propositional logic that uses the symbols 0 and 1, or True and False. Boolean algebra is concerned with binary variables and logic operations.
Boolean Algebra is fundamental in the development of digital electronics systems as they all use the concept of Boolean Algebra to execute commands. Apart from digital electronics this algebra also finds its application in Set Theory, Statistics, and other branches of mathematics.
What is Boolean Algebra?
Boolean Algebra is a branch of algebra that deals with boolean values—true and false. It is fundamental to digital logic design and computer science, providing a mathematical framework for describing logical operations and expressions
Boolean Algebra Operations
Various operations are used in Boolean algebra but the basic operations that form the base of Boolean Algebra are.
- Negationor NOT Operation
- Conjunctionor AND Operation
- Disjunctionor OR Operation
Operator | Symbol | Precedence |
|---|---|---|
NOT | ' (or) ⇁ | First |
AND | . (or) ∧ | Second |
OR | + (or) ∨ | Third |
We can easily define these operations using two Boolean variables.
Let's take two Boolean variables A and B that can have any of the two values 0 or 1, i.e. they can be either OFF or ON. Then these operations are explained as,
Negation or NOT Operation
Using the NOT operation reverse the value of the Boolean variable from 0 to 1 or vice-versa. This can be understood as:
- If A = 1, then using NOT operation we have (A)' = 0
- If A = 0, then using the NOT operation we have (A)' = 1
- We also represent the negation operation as ~A, i.e if A = 1, ~A = 0
Conjuntion or AND Operation
Using the AND operation satisfies the condition if both the value of the individual variables are true and if any of the value is false then this operation gives the negative result. This can be understood as,
- If A = True, B = True, then A . B = True
- If A = True, B = False, Or A = false, B = True, then A . B = False
- If A = False, B = False, then A . B = False
Disjuntion / OR Operation
Using the OR operation satisfies the condition if any value of the individual variables is true, it only gives a negative result if both the values are false. This can be understood as,
- If A = True, B = True, then A + B = True
- If A = True, B = False, Or A = false, B = True, then A + B = True
- If A = False, B = False, then A + B = False
Now, let us discuss the important terminologies covered in Boolean algebra. Boolean Algebra: Boolean algebra is the branch of algebra that deals with logical operations and binary variables. Boolean Variables: A Boolean variable is defined as a variable or a symbol defined as a variable or a symbol, generally an alphabet that represents the logical quantities such as 0 or 1. Boolean Function: A Boolean function consists of binary variables, logical operators, constants such as 0 and 1, equal to the operator, and the parenthesis symbols. Literal: A literal may be a variable or a complement of a variable. Complement: The complement is defined as the inverse of a variable, which is represented by a bar over the variable.Boolean Algebra Terminologies
Boolean Algebra Table
Given Below is the Expression for the Boolean Algebra
| Operation | Symbol | Definition |
|---|---|---|
| AND Operation | ⋅ or ∧ | Returns true only if both inputs are true. |
| OR Operation | + or ∨ | Returns true if at least one input is true. |
| NOT Operation | ¬ or ∼ | Reverses the input. |
| XOR Operation | ⊕ | Returns true if exactly one input is true. |
| NAND Operation | ↓ | Returns false only if both inputs are true. |
| NOR Operation | ↑ | Returns false if at least one input is true. |
| XNOR Operation | ↔ | Returns true if both inputs are equal. |
Truth Tables in Boolean Algebra
A truth table represents all the combinations of input values and outputs in a tabular manner. All the possibilities of the input and output are shown in it and hence the name truth table. In logic problems, truth tables are commonly used to represent various cases. T or 1 denotes 'True' & F or 0 denotes 'False' in the truth table.
Example: Draw the truth table of the conditions A + B and A.B where A and b are boolean variables.
Solution:
The required Truth Table is,
| A | B | X = A + B | Y = A.B |
|---|---|---|---|
| T | T | T | T |
| T | F | T | F |
| F | T | T | F |
| F | F | F | F |
Boolean Algebra Rules
In Boolean Algebra there are different fundamental rules for logical expression.
- Binary Representation: In Boolean Algebra the variables can have only two values either 0 or 1 where 0 represents Low and 1 represents high. These variables represents logical states of the system.
- Complement Representation: The complement of the variables is represented by (¬) or (') over the variable. This indicates logical negation or inversion of the variable's value. So Complement of variable A can be represented by
,if the value of A=0 then its complement is 1. - OR Operation: The OR operation is represented by (+) between the Variables. OR operation returns true if at least one of the operands is true. For Examples let us take three variables A,B,C the OR operation can be represented as A + B + C.
- AND Operation: The AND Operation is denoted by (.) between the Variables. AND operation returns true only if all the operands are true. For Examples let us take three variables A,B,C the AND operation can be represented A.B.C or ABC.