What is Boolean Algebra?
Boolean algebra is a logical mathematics in which variables
are represented by using the letters of the English alphabet and the value of a
variable can be either TRUE (1) or FALSE (0).
In simple words, Boolean algebra is a mathematics that deals
with the analysis and synthesis of logical operations.
Boolean algebra is also referred to as switching algebra or logical algebra. Boolean algebra is a mathematics that is based on the binary number system. Therefore, it is also called the mathematics of binary numbers.
Boolean algebra is the mathematics that is used to analyze
and implement logic circuits or switching circuits or gate circuits or digital
circuits.
Terminology of Boolean Algebra
(1). Literal:
A literal is a logical variable either in complemented or
un-complemented form.
(2). Binary Variable:
A variable that can have only two values, namely TRUE or
HIGH or ON or 1 or FALSE or LOW or OFF or 0 is called a binary variable. It is
also called a Boolean variable.
(3). Boolean Function:
A Boolean function is a logical expression consisting of
binary variables, logical operators, and constants is called a Boolean function
or Boolean expression.
(4). Complement:
The negation or inverse of a binary variable is called its
complement. It is usually denoted by a bar over the variable. For example, (`\overline{A}`) is a complement of
variable A. Sometimes, it is also denoted as A'.
(5). Truth Table:
A truth table is a table that provides information about
the outputs of a logical expression for all possible combinations of input
variables.
Boolean Algebra Operations
In Boolean algebra, there are three basic logic operations
namely, AND operation, OR operation, and NOT operation. In addition to these
three basic operations, there are four derived operations as well, namely NAND
operation, NOR operation, XOR operation, and XNOR operation.
Let’s discuss each type of logic operation in Boolean
algebra in detail.
(1). AND Operation:
AND operation is a logical multiplication. It is performed
by the AND gate. The AND operation can be represented by a dot (.) or
intersection (∩) or
caret (^).
(2). OR Operation:
AND operation is a logical addition. It is performed by the
OR gate. The OR operation can be represented by a plus (+) or union (∪).
(3). NOT Operation:
NOT operation is a Boolean operation that performs inversion
or complementation of a variable. NOT operation is performed by the NOT gate.
It is a unary operation, as it takes only one operand. NOT operation is
represented by a bar ‘-’ over the variable or a prime (').
(4). NAND Operation:
NAND operation is a combination of two basic operations
namely, AND operation and NOT operation. Thus, it is given by,
NAND Operation = AND
Operation + NOT Operation
NAND operation is performed by the NAND Gate or a
combination of the AND gate and NOT gate.
(5). NOR Operation:
NOR operation is a combination of OR operation and NOT
operation, i.e.
NOR Operation = OR
Operation + NOT Operation
It is performed by the NOR gate or a combination of the OR gate
and NOT gate.
(6). XOR Operation:
XOR operation, also called Exclusive OR operation or Ex-OR
operation, is a binary operation performed by a derived gate called XOR gate.
It is denoted by ⊕. XOR operation is also known as the modulo-2 addition.
(7). XNOR Operation:
XNOR operation is also a binary operation performed on two
variables. It is performed by using a logic gate named XNOR gate. It is also
known as Exclusive NOR operation or Ex-NOR operation. It is denoted by the symbol
⊙.
XNOR operation is also known as modulo-2
multiplication.
Laws of Boolean Algebra
The following table lists the various types of laws of
Boolean algebra:
S.
No. |
Boolean
Law |
Expression |
|||||
1. |
AND Law |
0 . 0 = 0 |
0 . 1 = 0 |
1 . 0 = 0 |
1 . 1 = 1 |
||
2. |
OR Law |
0 + 0 = 0 |
0 + 1 = 1 |
1 + 0 = 1 |
1 + 1 = 1 |
||
3. |
NOT Law |
0’ = 1 |
1’ = 0 |
`\overline{overline{A} }`= A |
|||
4. |
Commutative Law |
A + B = B + A |
A . B = B . A |
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5. |
Associative Law |
(A + B) + C = A + (B + C) |
(A . B) . C = A . (B . C) |
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6. |
Distributive Law |
A(B + C) = AB + AC |
A + BC = (A + B)(A + C) |
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Conclusion
In conclusion, Boolean algebra is a logical mathematics used
in digital electronics to analyze and synthesize digital logic circuits. It uses
variables that can have only two values i.e., TRUE or FALSE.
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