World Class Education in Chitwan
Boolean
Algebra
- Madhav Adhikari
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Introduction to Boolean Algebra
Boolean Algebra is used to analyze and simplify the digital (logic) circuits. It uses only the
binary numbers i.e. 0 and 1. It is also called as Binary Algebra or logical Algebra. Boolean
algebra was invented by George Boole in 1854.
It is used to analyze and simplify the digital or logical circuits. It only uses binary numbers
and is also called binary algebra or Logical Algebra.
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Differences between Ordinary Algebra and Boolean Algebra
• Boolean algebra does not have operations equivalent to division and subtraction.
• Ordinary algebra deals with real numbers, which contains an infinite number of
elements (1, 2, 3…). But Boolean algebra has only a finite set of elements. That is, it
deals with only two elements 0 and 1.
• Operations like addition, subtraction, division and so on are not used in Boolean
algebra. Here, the basic operations used are AND, OR, NOT, NAND, etc.
• In Boolean algebra, there are no coefficients or exponents, i.e. A+A= A and A.A= A
• The distributive law [(A +B) . (A+C)= A + (B.C)] does not hold on ordinary algebra.
• Unlike in ordinary algebra, there are several graphical methods of representing
Boolean expression.
Note: Boolean algebra has an application in binary things either 0 or 1, a fundamental
feature of modern digital electronic computers.
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Terms in Boolean Algebra
Boolean values
The values used in Boolean algebra are called Boolean values. They can only be either
true or false which can be represented by 1 and 0.
Boolean Variable
A variable with its value i.e. either true or false ( 1 or 0) is called a Boolean variable. It is
also known as logical variable. The variable can be denoted by A,B,C,X,Y and so on.
Boolean Expression
A Boolean expression is a statement using Boolean operators that expresses a condition
that is either true or false. A Boolean expression always produces a Boolean value. A
Boolean expression is composed of a combination of the Boolean constants (True or
False), Boolean variables and logical connectives. Each Boolean expression represents a
Boolean function.
Example: A.B′.C , A+B+C are Boolean expression.
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Terms in Boolean Algebra
Logic Function (Boolean Function)
A Logic function is an expression formed by binary variables, binary operators OR, AND,
unary operator NOT, parenthesis, and equal sign. For a given value of the variables, the
function can be either 0 or 1.
Example: Consider the Boolean function in an algebraic expression: F = X.Y.Z’+X.Y
Here, X, Y and Z are Boolean variables. The right hand side of the above equation is
known as expression.
Each occurrence of a variable or its complement in an expression is called literal. In the
above expression, there are three variables (X, Y and Z) and five literals (X, Y, Z’, X, and Y).
Logical Circuit
The high and low level voltage electrical signals are produced by using transistors or ICs
or LSIs (known as logical elements). The circuit composed of logical elements for a
specific operation is called logical circuit.
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Terms in Boolean Algebra
Truth Table
A table that represents the input-output relationship of the binary variables for each
logical gate is called a truth table. It shows the relationship between all inputs and output
in tabular form. Thus a truth table is a table representing the results of the logical
operations on all possible combinations of logical values. In another words, we can say
that a truth table is a useful tool for describing the terminal behavior of a logic network.
The number of rows in a truth table is equal to 2n (where n= no. of variables). For
example, for 3 input variables, no. of rows will be 23= 8.
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EXAMPLE:
No of inputs(n) = 3
No of inputs(n) = 2
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Boolean operation
Operator : An operator is a special symbol that indicates the operation to be carried out
between the operands.
Operands: They are the variables, constants or values on which operation is performed.
Operation: An operation is an action to be carried out upon the operands.
In Boolean Algebra there are three basic operations. They are:
• AND operator
• OR operator
• NOT operator
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Boolean operation
AND operation:
AND operator is also known as logical multiplication. It is carried out by dot(.) operator or
simply by AND. It generates true output if all inputs are true. Otherwise, the output will
be false.
The logical equation of AND operation is written as C = A.B or C = A AND B.
The truth table of AND operation is as below:
Inputs
Output
A
B
C = A.B
False
False
False
False
True
False
True
False
False
True
True
True
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Boolean operation
OR operation:
OR operator is also known as logical addition. It is carried out by dot(+) operator or
simply by OR. It generates true output if at least one input is true. Otherwise, the output
will be false.
The logical equation of OR operation is written as C = A+B or C = A OR B.
The truth table of OR operation is as below:
Inputs
Output
A
B
C = A+B
False
False
False
False
True
True
True
False
True
True
True
True
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Boolean operation
NOT operation:
NOT operator is also known as logical complement. It is carried out by prime(‘) or bar( )̅
operator or simply by NOT. This operator takes only one input. It generates true output if
the input is false. Otherwise, the output will be false.
The logical equation of NOT operation is written as C = A’.
The truth table of NOT operation is as below:
Inputs
Output
A
C = A’
False
True
True
False
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Logic Gate
A logic gate is an electronic circuit that operates on one or more input signals to
produce an output signal. The logic gate is used for binary operation and is the basic
component of a digital computer. It is embodied in the Integrated Circuit (IC). Each gate
has a specific function and graphical symbol.
It produces signals of binary 1 or 0 (ON or OFF) when input logic requirements are
satisfied. Each logic gate has a unique graphical symbol and its operation can be
represented in terms of Boolean expression.
In digital computers there are three basic logic gates: AND, OR and NOT.
Beside these, there are other gates which are derived rom these three basic gates and
they are: NAND gate, NOR gate, X-OR gate ( Exclusive OR gate) and X-NOR gate (Exclusive
NOR gate gate).
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AND Gate
AND gate is an electronic circuit, which produces true value(1) output when all inputs are
true. Otherwise, the output will be false (0).
The output is equal to the product of the logic inputs. It can have two or more inputs and
produces a single output.
It is denoted by a dot (“.”) and is also referred as arithmetic multiplication.
Its logic function is: F = A . B Where ‘.’ stands for AND operation, A and B are inputs and F is output.
Its logic or graphical symbol is:
Its truth table is:
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OR Gate
OR gate is an electronic circuit, which produces true (1) output when any one of the input
is true (1). If all inputs are false (0), then the output will also be false (0). The output is
equal to the sum of the logic inputs. It has two or more inputs and produces a single
output.
It is denoted by plus sign(“+”) and referred as arithmetic addition.
Its logic function is: F = A + B Where ‘+’ stands for OR operation, A and B are inputs and F is output.
Its logic or graphical symbol is:
Its truth table is:
Inputs
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Output
A
B
C = A+B
0
0
0
0
1
1
1
0
1
1
1
1
© Madhav Adhikari
NOT Gate
NOT gate is an electronic circuit whose output is the complement of the input. It is also
called an inverter. If we provide true input (1) to this gate, it will produce false output (0)
& vice-versa. It takes only one input and gives the output.
It is denoted by prime sign (‘).
Its logic function is: F = A’
Its logic or graphical symbol is:
Its truth table is:
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NAND Gate
The NAND gate is the combination of AND and NOT gate. This electronic gate produces
False (0) output, when all inputs are true (1), otherwise, the output will be true (1). It is
the complement of AND gate. It has two or more inputs and produces a single output.
Its logic function is: F = (A.B)‘
Its logic or graphical symbol is:
Its truth table is:
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NOR Gate
NOR gate is the combination of the OR gate and NOT gate. This electronic gate produces
true (1) output when all inputs are false (0) otherwise, the output will be false (0). It is
the complement of the OR gate. It has two or more inputs and produces a single output.
Its logic function is: F = (A+B)‘
Its logic or graphical symbol is:
Its truth table is:
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Exclusive-OR (X-OR) Gate
It is an electric circuit used to perform either / or operation. It can also have two or more
inputs which produces single output. The XOR gate produces true output (1) when the
inputs are at opposite logic levels or inputs are not same. It is also known as even parity
generator.
Its logic function is: F = A ⊕ B = A‘.B + A.B‘
Its logic or graphical symbol is:
Its truth table is:
Inputs
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Output
A
B
C=A⊕B
0
0
0
0
1
1
1
0
1
1
1
0
9/4/2020
© Madhav Adhikari
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Exclusive-NOR (X-NOR) Gate
It performs logical complement of X-OR operator. It also have two or more inputs which
produces single output. The XOR gate produces true output (1) when both input are at
same logic level else it gives false output(0).
Its logic function is: F = (A ⊕ B)‘ = A.B + A‘.B‘
Its logic or graphical symbol is:
Its truth table is:
Inputs
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Output
A
B
C = (A ⊕ B)’
0
0
1
0
1
0
1
0
0
1
1
1
© Madhav Adhikari
9/4/2020
Laws of Boolean algebra
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