Boolean algebra is a logical system used to simplify binary expressions. It uses binary variables that can have a value of 1 or 0, and logical operators like AND, OR and NOT. Boolean expressions can be represented in truth tables and minimized using Karnaugh maps or algebraic manipulation. Key concepts include minterms, maxterms, sum of products form, and product of sums form. Boolean algebra provides a standard way to simplify logical statements and circuits into their essential components.

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BOOLEAN ALGEBRA
1 P0INTS TO REMEMBER
1 .BOOLEAN ALGEBRA: Is a two state algebra or algebra of logics.It is also called
Switching Algebra. It is based on Binary number system and uses the numeric constants
0 and 1.
2. BINARY DECISION: The decision which results into either ‘ TRUE ‘ or ‘FALSE’
where TRUE stands for 1 and FALSE stands for 0.
3. LOGICAL VARIABLES(Boolean Variable or Binary valued Variable):The
variables which can stores values either TRUE or FALSE OR ‘0’ or ‘1’.
4 BINARY OPERATIONS :Is an operation in which for a set of variables result is the
values i.e 0 or 1.
5 BOOLEAN OPERATORS:: In Boolean Operation ,operators used are of three types:
a) NOT- It is a Unary Operator.i.e it operateson single variable and operation
performed by it is known as Complementation or Negation .Its symbol is “¯”or
“ ’ ”.e.g. A’ or Ā
b)AND – It is a Binary Operator . It operates on two variables and operation
.
performed by it is known as Logical Multiplication . Its Symbol is “ ” Or “ ˆ ”.
.
e.g. A B or Aˆ B.
c) OR- It is a Binary Operator . It operates on two variables and operation performed
by it is known as Logical Addition . Its symbol is “ +” or “ˇ”.
e.g. . A + B or A ˇ B.
6. TRUTH TABLE – A truth table is a table that describes the behaviour of a logic
gate.It shows all input and output possibilities for logical variables or statements.The
input patterns are written in Binary Progression.
7.TAUTOLOGY- If the result of a logical statement or exprssion is always true or ‘1’,
it is known as Tautology.
8. FALLACY-- If the result of a logical statement or exprssion is always False or ‘0’,
it is known as Fallacy.
9 .LOGIC GATES-A logic gate performs a logical operation on one or more logic
inputs and produces a single logic output.
Boolean algebra
Type Distinctive shape Truth table
between A & B
INPUT OUTPUT
A B A AND B
AND A ..B
0 0 0
0 1 0
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1 0 0
1 1 1
INPUT OUTPUT
A B A OR B
0 0 0
OR A+B 0 1 1
1 0 1
1 1 1
INPUT OUTPUT
Ā A NOT A
NOT 0 1
1 0
INPUT OUTPUT
A NAND
A B
B
NAND 0 0 1
0 1 1
1 0 1
1 1 0
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INPUT OUTPUT
A B A NOR B
0 0 1
NOR
0 1 0
1 0 0
1 1 0
INPUT OUTPUT
A B A XOR B
0 0 0
XOR 0 1 1
1 0 1
1 1 0
INPUT OUTPUT
A XNOR
A B
B
0 0 1
XNOR
0 1 0
1 0 0
1 1 1
BOOLEAN POSTULATES
 P1: X = 0 or X = 1
 P2: 0 . 0 = 0
 P3: 1 + 1 = 1
 P4: 0 + 0 = 0
 P5: 1 . 1 = 1
 P6: 1 . 0 = 0 . 1 = 0
 P7: 1 + 0 = 0 + 1 = 1
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LAWS OF BOOLEAN ALGEBRA
T1 : Commutative Law
(a) A + B = B + A
(b) A B = B A
T2 : Associate Law
(a) (A + B) + C = A + (B + C)
(b) (A B) C = A (B C)
T3 : Distributive Law
(a) A (B + C) = A B + A C
(b) A + (B C) = (A + B) (A + C)
T4 : Identity Law
(a) A + A = A
(b) A A = A
T5 :
(a)
(b)
T6 : Redundance Law
(a) A + A B = A
(b) A (A + B) = A
T7 :
(a) 0 + A = A
(b) 0 A = 0
T8 :
(a) 1 + A = 1
(b) 1 A = A
T9 :
(a)
(b)
T10 :
(a)
(b)
T11 : De Morgan's Theorem
(a)
(b)
Examples
Prove T10 : (a)
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S(1) Algebraically:
(2) Using the truth table:
Using the laws given above, complicated expressions can be simplified.
Problems
(a) Prove T10(b).
(b) Copy or print out the truth table below and use it to prove T11: (a) and (b).
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(b)
DEMORGAN THEOREM
(a) Proof of (A + B)' = A' . B',
These can be proved by the use of truth tables 1(a) & 1(b)
table 1(a)
A B A+B (A+B)'
0 0 0 1
0 1 1 0
1 0 1 0
1 1 1 0
table 1(b)
A B A' B' A'.B'
0 0 1 1 1
0 1 1 0 0
1 0 0 1 0
1 1 0 0 0
The two truth tables are identical, and so the two expressions are identical.
(b)Proof of (A.B) = A' + B'
(A.B) = A' + B', These can be proved by the use of truth tables 2(a) & 2(b)
table 2(a)
A B A.B (A.B)'
0 0 0 1
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0 1 0 1
1 0 0 1
1 1 1 0
table2(b)
A B A' B' A'+B'
0 0 1 1 1
0 1 1 0 1
1 0 0 1 1
1 1 0 0 0
Canonical form: standard form for a Boolean expression
provides a unique algebraic signature
Minterms and Maxterms
Any boolean expression may be expressed in terms of either minterms or
maxterms.
A literal is a single variable within a term which may or may not be
complemented. For an expression with N variables, minterms and maxterms are
defined as follows :
A minterm is the product of N distinct literals where each literal occurs exactly
once
A maxterm is the sum of N distinct literals where each literal occurs exactly once
For a two-variable expression, the minterms and maxterms are as follows
X Y Minterm Maxterm
0 0 X'.Y' X+Y
0 1 X'.Y X+Y'
1 0 X.Y' X'+Y
1 1 X.Y X'+Y'
For a three-variable expression, the minterms and maxterms are as follows
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X Y Z Minterm Maxterm
0 0 0 X'.Y'.Z' X+Y+Z
0 0 1 X'.Y'.Z X+Y+Z'
0 1 0 X'.Y.Z' X+Y'+Z
0 1 1 X'.Y.Z X+Y'+Z'
1 0 0 X.Y'.Z' X'+Y+Z
1 0 1 X.Y'.Z X'+Y+Z'
1 1 0 X.Y.Z' X'+Y'+Z
1 1 1 X.Y.Z X'+Y'+Z'
Sum Of Products (SOP)
The Sum of Products form represents an expression as a sum of
minterms.To derive the Sum of Products form from a truth table, OR together all
of the minterms which give a value of 1.
Example – SOP
Consider the truth table
X Y F Minterm
0 0 0 X'.Y'
0 1 0 X'Y
1 0 1 X.Y'
1 1 1 X.Y
Here SOP is f(X.Y) = X.Y' + X.Y
Product Of Sum (POS)
The Product of Sums form represents an expression as a product of maxterms.
To derive the Product of Sums form from a truth table, AND together all of the
maxterms which give a value of 0.
Example – POS
Consider the truth table from the previous example.
X Y F Maxterm
0 0 1 X+Y
0 1 0 X+Y'
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1 0 1 X'+Y
1 1 1 X'+Y'
Here POS is F(X,Y) = (X+Y')
Minimisation of Boolean Functions
In mathematics expressions are simplified to understand and easier to write
down, they are also less prone to error.
Minimisation can be achieved by a following methods:
1)Algebraic Manipulation of Boolean Expressions.
2)Karnaugh Maps
Algebraic Manipulation of Boolean Expressions
This is an approach where you can transform one boolean expression into an
equivalent expression by applying Boolean Theorems
Karnaugh Maps
 K-Maps are a convenient way to simplify Boolean Expressions.
 They can be used for up to 4 or 5 variables.
 They are a visual representation of a truth table.
 Expression are most commonly expressed in sum of products form.
.
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