boolean algebra class 11

1. CLASS-11
Chapter -3
BOOLEAN LOGIC
Prepared by
Ms. Ragavi.v(M.Sc..,(software
systems))

2. Fundamentals of Digital
Electronics - Logic Gates
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3. Logic Circuits
 A collection of individual logic gates connect with
each other and produce a logic design known as a
Logic Circuit
 The following are the types of logic circuits:
 Decision making
 Memory
 A gate has two or more binary inputs and single output.
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4. Basic Logic Gates
 The following are the three basic gates:
 NOT
 AND
 OR
 Each logic gate performs a different logic function. You
can derive logical function or any Boolean or logic
expression by combining these three gates.
Fundamentals of Digital Electronics - Logic Gates 4

5. NOT Gate
 The simplest form of a digital logic circuit is the
inverter or the NOT gate
 It consists of one input and one output and the
input can only be binary numbers namely; 0 and 1
the truth table for NOT Gate:
Fundamentals of Digital Electronics - Logic Gates 5

6. AND Gate
 The AND gate is a logic circuit that has two or more inputs
and a single output
 The operation of the gate is such that the output of the gate
is a binary 1 if and only if all inputs are binary 1
 Similarly, if any one or more inputs are binary 0, the output
will be binary 0.
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7. OR Gate
 The OR gate is another basic logic gate
 Like the AND gate, it can have two or more inputs and a
single output
 The operation of OR gate is such that the output is a binary
1 if any one or all inputs are binary 1 and the output is
binary 0 only when all the inputs are binary 0
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8. NAND Gate
 The term NAND is a contraction of the expression
NOT-AND gate
 A NAND gate, is an AND gate followed by an
inverter
 The algebraic output expression of the NAND gate
is Y = A.B
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9. NOR Gate
 The term NOR is a contradiction of the expression
NOT-OR
 A NOR gate, is an OR gate followed by an inverter
 The algebraic output expression of the NOR gate is
Y = A + B
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10. EX-OR and EX-NOR Gates
 EX-OR and EX-NOR are digital logic circuits that may
use two or more inputs
 EX-NOR gate returns the output opposite to EX-OR
gate
 EX-OR and EX-NOR gates are also denoted by XOR
and XNOR respectively.
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11. EXOR Gate
 The Ex-OR (Exclusive- OR) gate returns high output with one of
two high inputs (but not with both high inputs or both low
inputs)
 For example, if both the inputs are binary 0 or 1, it will return the
output as 0. Similarly, if one input is binary 1 and another is
binary 0, the output will be 1 (high)
 The operation for the Ex-OR gate is denoted by encircled plus
symbol
 The Ex-OR operation is widely used in digital circuits.
 The algebraic output expression of the Ex-OR gate is Y = A ⊕ B =
Fundamentals of Digital Electronics - Logic Gates 11

12. EXNOR Gate
 The Ex-NOR (Exclusive- NOR) gate is a circuit that returns low
output with one of two high inputs (but not with both high inputs)
 For example, if both the inputs are binary 0 or 1, it will return the
output as 1. Similarly, if one input is binary 1 and another is binary 0,
the output will be 0 (low)
 The symbol for the Ex-NOR gate is denoted by encircled plus symbol
which inverts the binary values
 The algebraic output expression of the Ex-NOR gate is Y =
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13. Applications of Logic Gates
 The following are some of the applications of Logic
gates:
 Build complex systems that can be used to different fields
such as
 Genetic engineering,
 Nanotechnology,
 Industrial Fermentation,
 Metabolic engineering and
 Medicine
 Construct multiplexers, adders and multipliers.
 Perform several parallel logical operations
 Used for a simple house alarm or fire alarm or in the circuit
of automated machine manufacturing industry
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14. Boolean Laws
 Three of the basic laws of Boolean
algebra are the same as in ordinary
algebra; the commutative law, the
associative law and the distributive law
 The commutative law for addition and
multiplication of two variables is
written as,
A + B = B + A
 A . B = B . A
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15. Associative law
This law states that the order in which the
logic operations are performed is
irrelevant as their effect is the same.
The associative law for addition and
multiplication of three variables is
written as,
 (A + B) + C = A + (B + C)
 (A .B) . C = A. (B. C)
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16. Distributive law
Distributive law states the following
condition.
The distributive law for three variables
involves, both addition and multiplication
and is written as,
 A (B+ C) = A B + AC
 Note that while either '+' and „.‟ s can be
used freely.
The two cannot be mixed without
ambiguity in the absence of further rules.
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17. For example does
 A . B + C means (A . B) + C or A . (B+ C)?
These two form different values for A = O,
B = 1 and C = 1, because we have
 (A . B) + C = (0.1) + 1 = 1
 A . (B + C) = 0 . (1 + 1) = 0
 which are different. The rule which is
used is that „.‟ is always performed before
'+'. Thus X . Y + Z is (X.Y) + Z.
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18.  Basic Duality in Boolean Algebra:
 We state the duality theorem without proof. Starting
with a Boolean relation, we can derive another
Boolean relation by 1.
 Changing each OR (+) sign to an AND (.) sign 2.
 Changing each AND (.) sign to an OR (+) sign. 3.
 Complementary each 0 and 1 For instance
A +0= A The dual relation is A . 1 = A
 Also since A (B + C) = AB + AC by distributive law.
 Its dual relation is A + B C = (A + B) (A + C)
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19. De Morgan’s Theorem,

20. Theorems of Boolean Algebra
9) A = A
10) A + AB = A
11) A + AB = A + B
12) (A + B)(A + C) = A + BC
13) Commutative : A + B = B + A
AB = BA
14) Associative : A+(B+C) =(A+B) + C
A(BC) = (AB)C
15) Distributive : A(B+C) = AB +AC
(A+B)(C+D)=AC + AD + BC + BD

21. De Morgan’s Theorems
16) (X+Y) = X . Y
17) (X.Y) = X + Y
• Two most important theorems of Boolean
Algebra were contributed by De Morgan.
• Extremely useful in simplifying expression in
which product or sum of variables is inverted.
• The TWO theorems are :

22. Implications of De Morgan’s Theorem
(a) Equivalent circuit implied by theorem (16)
(b) Alternative symbol for the NOR function
(c) Truth table that illustrates DeMorgan’s Theorem
(a)
(b)
Input Output
X Y X+Y XY
0 0 1 1
0 1 0 0
1 0 0 0
1 1 0 0
(c)

23. Implications of De Morgan’s Theorem
(a) Equivalent circuit implied by theorem (17)
(b) Alternative symbol for the NAND function
(c) Truth table that illustrates DeMorgan’s Theorem
(a)
(b)
Input Output
X Y XY X+Y
0 0 1 1
0 1 1 1
1 0 1 1
1 1 0 0
(c)

24. De Morgan’s Theorem Conversion
Step 1: Change all ORs to ANDs and all ANDs to Ors
Step 2: Complement each individual variable
(short overbar)
Step 3: Complement the entire function (long overbars)
Step 4: Eliminate all groups of double overbars
Example : A . B A .B. C
= A + B = A + B + C
= A + B = A + B + C
= A + B = A + B + C
= A + B

25. De Morgan’s Theorem Conversion
ABC + ABC (A + B +C)D
= (A+B+C).(A+B+C) = (A.B.C)+D
= (A+B+C).(A+B+C) = (A.B.C)+D
= (A+B+C).(A+B+C) = (A.B.C)+D
= (A+B+C).(A+B+C) = (A.B.C)+D

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29. THANK YOU
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