Boolean Algebra in Computer Science

 
Unit 1
Unit 1
 
Computer Systems and Organisation
Computer Systems and Organisation
(CSO)
(CSO)
 
Developed by English Mathematician
Developed by English Mathematician
George Boole in between 1815 - 1864.
George Boole in between 1815 - 1864.
It is described as an algebra of logic or
It is described as an algebra of logic or
an algebra of two values i.e True or
an algebra of two values i.e True or
False.
False.
The term logic means a statement
The term logic means a statement
having binary decisions i.e True/Yes or
having binary decisions i.e True/Yes or
False/No.
False/No.
 
It is used to perform the logical
It is used to perform the logical
operations in digital computer.
operations in digital computer.
In digital computer True represent by ‘1’
In digital computer True represent by ‘1’
(high volt) and False represent by ‘0’ (low
(high volt) and False represent by ‘0’ (low
volt)
volt)
Logical operations are performed by
Logical operations are performed by
logical operators. The fundamental logical
logical operators. The fundamental logical
operators are:
operators are:
  
  
1.
1.
 
 
AND (conjunction)
AND (conjunction)
  
  
2.
2.
 
 
OR (disjunction)
OR (disjunction)
  
  
3.
3.
 
 
NOT (negation/complement)
NOT (negation/complement)
 
 
 
It performs logical multiplication and denoted
It performs logical multiplication and denoted
by (.) dot.
by (.) dot.
   
   
X
X
 
 
Y
Y
 
 
X.Y
X.Y
   
   
0
0
 
 
0
0
 
 
0
0
   
   
0
0
 
 
1
1
 
 
0
0
   
   
1
1
 
 
0
0
 
 
0
0
   
   
1
1
 
 
1
1
 
 
1
1
 
 
 
It performs logical addition and denoted
It performs logical addition and denoted
by (+) plus.
by (+) plus.
  
  
X
X
 
 
Y
Y
 
 
X+Y
X+Y
  
  
0
0
 
 
0
0
 
 
0
0
  
  
0
0
 
 
1
1
 
 
1
1
  
  
1
1
 
 
0
0
 
 
1
1
  
  
1
1
 
 
1
1
 
 
1
1
 
  
  
It performs logical negation and
It performs logical negation and
denoted by (-) bar. It operates on single
denoted by (-) bar. It operates on single
variable.
variable.
  
  
X
X
 
 
X
X
 
 
(means complement of x)
(means complement of x)
  
  
0
0
 
 
1
1
  
  
1
1
 
 
0
0
 
 
 
 
 
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=
=
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,
 
 
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.
 
 
 
 
 
 
 
 
 
 
 
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+
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Z
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a
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:
:
D
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X
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0
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2
2
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0
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3
3
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0
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4
4
1
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0
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5
5
1
1
0
0
1
1
0
0
1
1
6
6
1
1
1
1
0
0
1
1
1
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7
7
1
1
1
1
1
1
1
1
1
1
 
 
 
 
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.
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0
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.
.
 
1. Evaluate the following Boolean
1. Evaluate the following Boolean
expression using Truth Table.
expression using Truth Table.
(a) X’Y’+X’Y
(a) X’Y’+X’Y
  
  
(b) X’YZ’+XY’
(b) X’YZ’+XY’
(c) XY’(Z+YZ’)+Z’
(c) XY’(Z+YZ’)+Z’
 
2. Verify that P+(PQ)’ is a Tautology.
2. Verify that P+(PQ)’ is a Tautology.
3. Verify that (X+Y)’=X’Y’
3. Verify that (X+Y)’=X’Y’
 
   
   
Boolean Algebra applied in
Boolean Algebra applied in
computers electronic circuits. These
computers electronic circuits. These
circuits perform Boolean operations
circuits perform Boolean operations
and these are called logic circuits or
and these are called logic circuits or
logic gates.
logic gates.
 
 
 
 
 
A
A
 
 
g
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.
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1
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:
:
 
 
 
 
T
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.
 
 
 
 
T
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.
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.
 
 
 
 
 
 
T
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O
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T
 
 
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.
.
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.
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.
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r
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.
.
 
 
 
So while going out of the house you set
So while going out of the house you set
the "Alarm Switch" and if the burglar enters he
the "Alarm Switch" and if the burglar enters he
will set the "Person switch", and tada the alarm
will set the "Person switch", and tada the alarm
will ring.
will ring.
 
Electronic door will only open if it
Electronic door will only open if it
detects a person and the switch is set to
detects a person and the switch is set to
unlocked.
unlocked.
 
Microwave will only start if the start
Microwave will only start if the start
button is pressed and the door close
button is pressed and the door close
switch is closed.
switch is closed.
 
 
 
You would of course want your doorbell
You would of course want your doorbell
to ring when someone presses either the front
to ring when someone presses either the front
door switch or the back door switch..(nice)
door switch or the back door switch..(nice)
 
When the temperature falls below 20c
When the temperature falls below 20c
the Not gate will set on the central heating
the Not gate will set on the central heating
system (cool huh).
system (cool huh).
 
 
 
Known as a “universal” gate
Known as a “universal” gate
because ANY digital circuit can be
because ANY digital circuit can be
implemented with NAND gates alone.
implemented with NAND gates alone.
 
NAND
NAND
 
X
X
 
Y
Y
 
Z
Z
 
X  Y  Z
X  Y  Z
0  0  1
0  0  1
0  1  1
0  1  1
1  0  1
1  0  1
1  1  0
1  1  0
 
Z = ~(X & Y)
Z = ~(X & Y)
nand(Z,X,Y)
nand(Z,X,Y)
 
NOR
NOR
 
X
X
 
Y
Y
 
Z
Z
 
X  Y  Z
X  Y  Z
0  0  1
0  0  1
0  1  0
0  1  0
1  0  0
1  0  0
1  1  0
1  1  0
 
Z = ~(X | Y)
Z = ~(X | Y)
nor(Z,X,Y)
nor(Z,X,Y)
 
X Y  Z
X Y  Z
 
XOR
XOR
 
X
X
 
Y
Y
 
Z
Z
 
0 0  0
0 0  0
 
0 1  1
0 1  1
 
1 0  1
1 0  1
 
1 1  0
1 1  0
 
Z = X ^ Y
Z = X ^ Y
xor(Z,X,Y)
xor(Z,X,Y)
 
X Y  Z
X Y  Z
 
XNOR
XNOR
 
X
X
 
Y
Y
 
Z
Z
 
0 0  1
0 0  1
 
0 1  0
0 1  0
 
1 0  0
1 0  0
 
1 1  1
1 1  1
 
Z = ~(X ^ Y)
Z = ~(X ^ Y)
Z = X ~^ Y
Z = X ~^ Y
xnor(Z,X,Y)
xnor(Z,X,Y)
 
 
T1 : Properties of 0
T1 : Properties of 0
(a) 
(a) 
0 + A = A
0 + A = A
(b) 
(b) 
0 A = 0
0 A = 0
T2 : Properties of 1
T2 : Properties of 1
(a) 
(a) 
1 + A = 1
1 + A = 1
(b) 
(b) 
1 A = A
1 A = A
 
T3 : Commutative Law
T3 : Commutative Law
(a) 
(a) 
A + B = B + A
A + B = B + A
(b) 
(b) 
A B = B A
A B = B A
T4 : Associate Law
T4 : Associate Law
(a) 
(a) 
(A + B) + C = A + (B + C)
(A + B) + C = A + (B + C)
(b) 
(b) 
(A B) C = A (B C)
(A B) C = A (B C)
T5 : Distributive Law
T5 : Distributive Law
(a) 
(a) 
A (B + C) = A B + A C
A (B + C) = A B + A C
(b) 
(b) 
A + (B C) = (A + B) (A + C)
A + (B C) = (A + B) (A + C)
(c) A+A’B = A+B
(c) A+A’B = A+B
 
 
T6 : Indempotence (Identity ) Law
T6 : Indempotence (Identity ) Law
(a) 
(a) 
A + A = A
A + A = A
(b) 
(b) 
A A = A
A A = A
 
T7 : Absorption (Redundance) Law
T7 : Absorption (Redundance) Law
(a) 
(a) 
A + A B = A
A + A B = A
(b) 
(b) 
A (A + B) = A
A (A + B) = A
 
 
T8 : Complementary Law
T8 : Complementary Law
(a) X+X’=1
(a) X+X’=1
(b) X.X’=0
(b) X.X’=0
T9 : Involution
T9 : Involution
(a)  x’’   = x
(a)  x’’   = x
T10 : De Morgan's Theorem
T10 : De Morgan's Theorem
(a) (X+Y)’=X’.Y’
(a) (X+Y)’=X’.Y’
(b) (X.Y)’=X’+Y’
(b) (X.Y)’=X’+Y’
 
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Thank You
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Boolean algebra, developed by mathematician George Boole, is essential in computer systems for performing logical operations with True and False values represented as 1 and 0. It involves operators like AND, OR, and NOT, enabling digital computers to process information effectively. Truth tables aid in visualizing outcomes, and terms like tautology and fallacy describe the validity of expressions. Explore the significance and application of Boolean algebra in the realm of computer science.


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  1. XI Computer Science (083) Board : CBSE Unit 1 Computer Systems and Organisation (CSO) CHAPTER 03 BOOLEAN ALGEBRA

  2. INTRODUCTION

  3. INTRODUCTION Developed by English Mathematician George Boole in between 1815 - 1864. It is described as an algebra of logic or an algebra of two values i.e True or False. The term logic means a statement having binary decisions i.e True/Yes or False/No.

  4. APPLICATION OF BOOLEAN ALGEBRA

  5. APPLICATION OF BOOLEAN ALGEBRA It is used to perform the logical operations in digital computer. In digital computer True represent by 1 (high volt) and False represent by 0 (low volt) Logical operations are performed by logical operators. The fundamental logical operators are: 1. AND (conjunction) 2. OR (disjunction) 3. NOT (negation/complement)

  6. AND operator It performs logical multiplication and denoted by (.) dot. X Y X.Y 0 0 0 1 1 0 1 1 0 0 0 1

  7. OR operator It performs logical addition and denoted by (+) plus. X Y X+Y 0 0 0 0 1 1 1 0 1 1 1 1

  8. NOT operator denoted by (-) bar. It operates on single variable. X X (means complement of x) 0 1 1 0 It performs logical negation and

  9. Truth Table Truth table is a table that contains all possible values variables/statements in a Boolean expression. of logical No. of possible combination = 2n, where n=number of variables used in a Boolean expression.

  10. Truth Table The truth table for XY + Z is as follows: Dec X Y 0 0 0 1 0 0 2 0 1 3 0 1 4 1 0 5 1 0 6 1 1 7 1 1 Z 0 1 0 1 0 1 0 1 XY 0 0 0 0 0 0 1 1 XY+Z 0 1 0 1 0 1 1 1

  11. Tautology & Fallacy expression is always True or 1 is called Tautology. If the output of Boolean If the output of Boolean expression is always False or 0 is called Fallacy.

  12. Tautology & Fallacy

  13. Exercise 1. Evaluate the following Boolean expression using Truth Table. (a) X Y +X Y (c) XY (Z+YZ )+Z (b) X YZ +XY 2. Verify that P+(PQ) is a Tautology. 3. Verify that (X+Y) =X Y

  14. Implementation computers electronic circuits. These circuits perform Boolean operations and these are called logic circuits or logic gates. Boolean Algebra applied in

  15. Logic Gate

  16. Logic Gate operates on one or more signals and produce single output. Gates are digital circuits because the input and output signals are denoted by either 1(high voltage) or 0(low voltage). There are three basic gates and are: 1. AND gate A gate is an digital circuit which 2. OR gate 3. NOT gate

  17. AND gate

  18. AND gate The AND gate is an electronic circuit that gives a high output (1) only if all its inputs are high. AND gate takes two or more input signals and produce only one output signal. Input A 0 0 1 1 Input B 0 1 0 1 Output AB 0 0 0 1

  19. OR gate

  20. OR gate The OR gate is an electronic circuit that gives a high output (1) if one or more of its inputs are high. OR gate also takes two or more input signals and produce only one output signal. Input A 0 0 1 1 Input B 0 1 0 1 Output A+B 0 1 1 1

  21. NOT gate

  22. NOT gate The NOT gate is an electronic circuit that gives a high output (1) if its input is low . NOT gate takes only one input signal and produce only one output signal. The output of NOT gate is complement of its input. It is also called inverter. Input A Output A 0 1 1 0

  23. PRACTICAL APPLICATIONS OF LOGIC GATES

  24. PRACTICAL APPLICATIONS OF LOGIC GATES AND Gate So while going out of the house you set the "Alarm Switch" and if the burglar enters he will set the "Person switch", and tada the alarm will ring.

  25. PRACTICAL APPLICATIONS OF LOGIC GATES AND Gate Electronic door will only open if it detects a person and the switch is set to unlocked. Microwave will only start if the start button is pressed and the door close switch is closed.

  26. PRACTICAL APPLICATIONS OF LOGIC GATES OR Gate You would of course want your doorbell to ring when someone presses either the front door switch or the back door switch..(nice)

  27. PRACTICAL APPLICATIONS OF LOGIC GATES NOT Gate When the temperature falls below 20c the Not gate will set on the central heating system (cool huh).

  28. NAND, NOR XOR, XNOR GATES

  29. NAND Gate Known as a universal gate because ANY digital circuit can be implemented with NAND gates alone.

  30. NAND Gate NAND X Y Z 0 0 1 0 1 1 1 0 1 1 1 0 X Z Y Z = ~(X & Y) nand(Z,X,Y)

  31. NAND Gate F = X F = (X X) = X +X = X X X X Y X Y F X Y F = ((X Y) ) = (X +Y ) = X Y = X Y X X F = X+Y F = (X Y ) = X +Y = X+Y Y Y

  32. NOR Gate

  33. NOR Gate NOR X Y Z 0 0 1 0 1 0 1 0 0 1 1 0 X Y Z Z = ~(X | Y) nor(Z,X,Y)

  34. Exclusive-OR Gate

  35. Exclusive-OR Gate XOR X Y Z 0 0 0 0 1 1 1 0 1 1 1 0 X Y Z = X ^ Y xor(Z,X,Y) Z

  36. Exclusive-NOR Gate

  37. Exclusive-NOR Gate XNOR X Y Z 0 0 1 0 1 0 1 0 0 1 1 1 X Y Z = ~(X ^ Y) Z = X ~^ Y xnor(Z,X,Y) Z

  38. POWER CONSUMPTION OF SYSTEM

  39. Basic Theorem of Boolean Algebra T1 : Properties of 0 (a) 0 + A = A (b) 0 A = 0 T2 : Properties of 1 (a) 1 + A = 1 (b) 1 A = A

  40. Basic Theorem of Boolean Algebra T3 : Commutative Law (a) A + B = B + A (b) A B = B A T4 : Associate Law (a) (A + B) + C = A + (B + C) (b) (A B) C = A (B C) T5 : Distributive Law (a) A (B + C) = A B + A C (b) A + (B C) = (A + B) (A + C) (c) A+A B = A+B

  41. Basic Theorem of Boolean Algebra T6 : Indempotence (Identity ) Law (a) A + A = A (b) A A = A T7 : Absorption (Redundance) Law (a) A + A B = A (b) A (A + B) = A

  42. Basic Theorem of Boolean Algebra T8 : Complementary Law (a) X+X =1 (b) X.X =0 T9 : Involution (a) x = x T10 : De Morgan's Theorem (a) (X+Y) =X .Y (b) (X.Y) =X +Y

  43. De Morgan's Theorem

  44. De Morgan's Theorem 1 Theorem 1 A . B = A + B

  45. De Morgan's Theorem 1 Theorem 1 A . B = A + B

  46. De Morgan's Theorem 1 Theorem 1 A . B = A + B

  47. De Morgan's Theorem 2 Theorem 1 A + B = A . B

  48. De Morgan's Theorem 2 Theorem 2 A + B = A . B

  49. De Morgan's Theorem 2 Theorem 2 A + B = A . B

  50. De Morgan's Theorem 2 Theorem 2 A + B = A . B

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