Relationship Between Basic Operation of Boolean and Basic Logic Gate

1
Relationship Between Basic Operation of Boolean
and Basic Logic Gate

• The basic construction of a logical circuit is
  gates
• Gate is an electronic circuit that emits an
  output signal as a result of a simple Boolean
  operation on its inputs
• Logical function is presented through the
  combination of gates
• The basic gates used in digital logic is the same
  as the basic Boolean algebra operations (e.g.,
  AND, OR, NOT,)

2

• The package Truth Tables and Boolean Algebra set
  out the basic principles of logic.

the symbols, algebra signs and the truth table
for the gates
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Basic Theorems of Boolean Algebra
1. Identity Elements 2. Inverse
Elements

1 . A A A . A 0

0 A
A A A 1   3.
Idempotent Laws 4. Boundess Laws
A A A A 1 1 
A . A A
A . 0 0   5. Distributive Laws 6.
Order Exchange Laws A . (B C) A.B
A.C A . B B . A  A
(B . C) (AB) . (AC) A B B
A   7. Absorption Laws 8. Associative
Laws A (A . B) A A (B C) (A
B) C A . (A B) A A . (B . C) (A .
B) . C   9. Elimination Laws 10. De
Morgan Theorem ?
?????? ? ?
A (A . B) A B (A B)
A . B ?
????? ? ? A .
(A B) A . B (A . B) A B
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Exercise 1

• Apply De Morgan theorem to the following
  equations
• F V A L
• F A B C D
• Verify the following expressions
• S.T V.W R.S.T S.T V.W
• A.B A.C B.A A.B A.C

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Relationship Between Boolean Function and Logic
Circuit
Boolean function ? Q AB B (NOT A AND
B) OR B
Logic circuit
Q
B
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Relationship Between Boolean Function and Logic
Circuit

• Any Boolean function can be implemented in
  electronic form as a network of gates called
  logic circuit

A.B AB
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G A . (B C D)
G A . (B C D)
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Truth Table
9
Produce a truth table from the logic circuit
0 0 1 0 0
0 1 1 1 1
1 0 0 0 0
1 1 0 0 1
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Exercise 2

• Build a truth table for the following Boolean
  function

G A . (B C D)
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Karnaugh Map

• A graphical way of depicting the content of a
  truth table where the adjacent expressions differ
  by only one variable
• For the purposes simplification, the Karnaugh map
  is a convenient way of representing a Boolean
  function of a small number (up to four) of
  variables
• The map is an array of 2n squares, representing
  all possible combination of values of n binary
  variables
• Example 2 variables, A and B

B
B
B
1
0
A
A
A B A B
A B A B
00 01
10 11
0
A
1
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4 variables, A, B, C, D ? 24 16 squares
CD
AB
C D
C D
C D
C D
0000 0001
0100
1100
1000
A B
A B
A B
A B
13
00 01 11 10
AB

• List combinations in the order 00, 01, 11, 10

A B
A B
A B
A B
C
000 010 110 100
001 011 111 101
C
0 1
C
0 1
C
AB
C
000 001
010 011
110 111
100 101
A B
00 01 11 10
A B
A B
A B
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How to create Karnaugh Map
Truth Table
A B C F
0 0 0 1
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 0

1. Place 1 in the corresponding square

Karnaugh Map
BC
B C
B C
B C
B C
0 0 0 1 1 1 1 0
A
1 1
1 1
0
A
1
A
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Karnaugh Maps to Represent Boolean Functions
0 0 0 1 1 1 1 0
AB
A B
A B
A B
A B
1 1
16

• Group the adjacent squares
• Begin grouping square with 2n-1 for n variables
• e.g. 3 variables, A, B, and C
• 23-1 22 4
• 21 2
• 20 1

BC
B C
B C
B C
B C
0 0 0 1 1 1 1 0
A
1 1
1 1
0
A
1
A
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3 variables 23-1 22 4 22-1 21 2 21-1
20 1
BC
B C
B C
B C
B C
0 0 0 1 1 1 1 0
A
1
1 1 1 1
0
A
1
A
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4 variables, A, B, C, D ? 24-1 23 8
(maximum) 22 4 21 2 20 1 (minimum)
CD
AB
01
00
10
11
1 1
1
1
1 1 1
00
01
11
10

BD

F
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The following diagram illustrates some of the
possible pairs of values for which simplification
is possible
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Karnaugh Map
Boolean Function
Logic Circuit
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Exercise 3
Transform the following truth table to Karnaugh
Map and find the Boolean function