This document provides an overview of digital logic circuits and Boolean algebra. It defines basic logic operators like AND, OR, and NOT. It introduces Boolean algebra concepts such as Boolean variables, algebraic manipulation, and theorems. It also discusses Boolean functions using truth tables and logic circuits. Finally, it covers standard forms like Sum of Products and Product of Sums, logic gates, and De Morgan's theorem.
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B.sc cs-ii-u-1.5 digital logic circuits, digital component
1. Digital Logic Circuits, Digital
Component and Data
Representation
Course: B.Sc-CS-II
Subject: Computer Organization
And Architecture
Unit-1
1
2. Basic Definitions
• Binary Operators
– AND
z = x • y = x y z=1 if x=1 AND y=1
– OR
z = x + y z=1 if x=1 OR y=1
– NOT
z = x = x’ z=1 if x=0
• Boolean Algebra
– Binary Variables: only ‘0’ and ‘1’ values
– Algebraic Manipulation
3. Boolean Algebra Postulates
• Commutative Law
x • y = y • x x + y = y + x
• Identity Element
x • 1 = x x + 0 = x
• Complement
x • x’ = 0 x + x’ = 1
4. Boolean Algebra Theorems
• Duality
– The dual of a Boolean algebraic expression is obtained
by interchanging the AND and the OR operators and
replacing the 1’s by 0’s and the 0’s by 1’s.
– x • ( y + z ) = ( x • y ) + ( x • z )
– x + ( y • z ) = ( x + y ) • ( x + z )
• Theorem 1
– x • x = x x + x = x
• Theorem 2
– x • 0 = 0 x + 1 = 1
Applied to a
valid equation
produces a
valid equation
5. Boolean Algebra Theorems
• Theorem 3: Involution
– ( x’ )’ = x ( x ) = x
• Theorem 4: Associative & Distributive
– ( x • y ) • z = x • ( y • z ) ( x + y ) + z = x + ( y + z )
– x • ( y + z ) = ( x • y ) + ( x • z )
x + ( y • z ) = ( x + y ) • ( x + z )
• Theorem 5: De Morgan
– ( x • y )’ = x’+ y’ ( x + y )’ = x’• y’
– ( x • y ) = x + y ( x + y ) = x • y
• Theorem 6: Absorption
– x • ( x + y ) = x x + ( x • y ) = x
6. Boolean Functions[1]
• Boolean Expression
Example: F =
x + y’ z
• Truth Table
All possible combinations
of input variables
• Logic Circuit
x y z F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1x
y
z
F
7. Algebraic Manipulation
• Literal:
A single variable within a term that may be
complemented or not.
• Use Boolean Algebra to simplify Boolean
functions to produce simpler circuits
Example: Simplify to a minimum number of literals
F = x + x’y ( 3 Literals)
= x + ( x’y )
= ( x + x’ ) ( x + y )
= ( 1 ) ( x + y ) = x + y ( 2 Literals)
Distributive law (+ over
•)
8. Complement of a Function
• DeMorgan’s Theorm
• Duality & Literal Complement
CBAF
CBAF
CBAF
CBAF
CBAF
CBAF
9. Canonical Forms[1]
• Minterm
– Product (AND function)
– Contains all variables
– Evaluates to ‘1’ for a
specific combination
Example
A = 0 A B C
B = 0 (0) • (0) • (0)
C = 0
1 • 1 • 1 = 1
A B C Minterm
0 0 0 0 m0
1 0 0 1 m1
2 0 1 0 m2
3 0 1 1 m3
4 1 0 0 m4
5 1 0 1 m5
6 1 1 0 m6
7 1 1 1 m7
10. Canonical Forms[1]
• Maxterm
– Sum (OR function)
– Contains all variables
– Evaluates to ‘0’ for a
specific combination
Example
A = 1 A B C
B = 1 (1) + (1) + (1)
C = 1
0 + 0 + 0 = 0
A B C Maxterm
0 0 0 0 M0
1 0 0 1 M1
2 0 1 0 M2
3 0 1 1 M3
4 1 0 0 M4
5 1 0 1 M5
6 1 1 0 M6
7 1 1 1 M7
11. Canonical Forms[1]
• Truth Table to Boolean Function
CBAF CBA CBA ABCA B C F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1
12. Canonical Forms
• Sum of Minterms
• Product of Maxterms
ABCCBACBACBAF
7541 mmmmF
)7,5,4,1(F
CABBCACBACBAF
CABBCACBACBAF
CABBCACBACBAF
))()()(( CBACBACBACBAF
6320 MMMMF
(0,2,3,6)F
13. Standard Forms
• Sum of Products (SOP)
ABCCBACBACBAF BA
BA
CCBA
)1(
)(
AC
BBAC
)(
CB
AACB
)(
)()()( BBACCCBAAACBF
ACBACBF
14. Standard Forms
• Product of Sums (POS)
CABBCACBACBAF
)( CCBA
)( AACB
)( BBCA
)()()( AACBCCBABBCAF
CBBACAF
))()(( CBBACAF
15. Two-Level Implementations
• Sum of Products (SOP)
• Product of Sums (POS)
B’
C
FB’
A
A
C
A
C
FB’
A
B’
C
ACBACBF
))()(( CBBACAF
16. Logic Operators
• AND
• NAND (Not AND)
x
y
x • y
x
y
x • y
x y AND
0 0 0
0 1 0
1 0 0
1 1 1
x y NAND
0 0 1
0 1 1
1 0 1
1 1 0
17. Logic Operators
• OR
• NOR (Not OR)
x
y
x + y
x
y
x + y
x y OR
0 0 0
0 1 1
1 0 1
1 1 1
x y NOR
0 0 1
0 1 0
1 0 0
1 1 0
18. Logic Operators
• XOR (Exclusive-OR)
• XNOR (Exclusive-NOR)
(Equivalence)
x
y
x Å y
x y + x y
x
y
x Å y
x y
x y + x y
x y XOR
0 0 0
0 1 1
1 0 1
1 1 0
x y XNOR
0 0 1
0 1 0
1 0 0
1 1 1
21. De Morgan’s Theorem on Gates
• AND Gate
– F = x • y F = (x • y) F = x + y
• OR Gate
– F = x + y F = (x + y) F = x • y
Change the “Shape” and “bubble” all lines
22. References
1. Computer Organization and Architecture, Designing
for performance by William Stallings, Prentice Hall
of India.
2. Modern Computer Architecture, by Morris Mano,
Prentice Hall of India.
3. Computer Architecture and Organization by John P.
Hayes, McGraw Hill Publishing Company.
4. Computer Organization by V. Carl Hamacher,
Zvonko G. Vranesic, Safwat G. Zaky, McGraw Hill
Publishing Company.