LOGIC GATES

1. 5/29/2020UET 1
BOOLEAN ALGEBRA &
LOGIC GATES

2. COMPLEMENT OF A FUNCTION
• THE COMPLEMENT OF A FUNCTION F IS F′
• OBTAINED BY CHANGING EVERY 0 TO 1 AND EVERY 1 TO 0
• DERIVED BY USING DE-MORGAN’S LAWS
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Digital Logic
Design

3. DE – MORGAN’S LAWS
1. A + B = A . B
2. A . B = A + B
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Digital Logic
Design

4. EXAMPLE
F = X′.Y.Z′ + X′.Y′.Z
COMPLEMENTING BOTH SIDES:
F = X′.Y.Z′ + X′.Y′.Z
APPLYING DE-MORGAN’S LAWS:
= (X′.Y.Z′) . (X′.Y′.Z)
= (X′′ + Y′ + Z′′) . (X′′ + Y′′ + Z′)
= (X + Y′ + Z) . (X + Y + Z′)
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Digital Logic
Design

5. CIRCUIT DIAGRAM
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Digital Logic
Design
.
.
.
.Logic
Circuit
Inputs Outputs

6. EXAMPLES
FIND THE COMPLEMENT AND REDUCE THEM TO A
MINIMUM NO. OF LITERALS:
 F1 = X + X.Y
 F2 = AB′ + C′D′
 G = X.Y + X.Y′ + Y′.Z
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Digital Logic
Design

7. HOMEWORK
• EXAMPLE 2.2 – 2.3
• PROBLEM 2.7
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Digital Logic
Design

8. READING MATERIAL *EXTRA’S
APPLICATIONS OF BOOLEAN FUNCTIONS –
CIRCUIT CHIP DESIGN
QUESTION
A FIRE SPRINKLER SYSTEM SHOULD SPRAY WATER IF
HIGH HEAT
IS SENSED AND THE SYSTEM IS SET TO ENABLED.
ANSWER
LET BOOLEAN VARIABLE H REPRESENT “HIGH HEAT IS
SENSED,” E
REPRESENT “ENABLED,” AND F REPRESENT “SPRAYING
WATER.”
THEN AN EQUATION IS:
F = H AND E
SYMBOLICALLY, F = H . E
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Digital Logic
Design
h
e
F

9. SEAT BELT WARNING LIGHT
SYSTEM
DESIGN CIRCUIT FOR WARNING LIGHT
• SENSORS
S=1: SEAT BELT FASTENED
K=1: KEY INSERTED
P=1: PERSON IN SEAT
CAPTURE BOOLEAN EQUATION
PERSON IN SEAT, AND SEAT BELT NOT
FASTENED, AND KEY INSERTED
• CONVERT EQUATION TO CIRCUIT
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Digital Logic
Design
w = p AND NOT (s) AND k

10. REPRESENTATION OF BOOLEAN
EXPRESSIONS
• CANONICAL FORMS (NOT SIMPLIFIED)
• STANDARD FORMS (SIMPLIFIED)
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11. CANONICAL FORMS
• DIRECTLY OBTAINED FROM THE TRUTH TABLE
• NOT SIMPLIFIED
• TWO TYPES:
• SUM OF MINTERMS
• PRODUCT OF MAXTERMS
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12. MINTERMS
• OBTAINED BY AND-ING ALL VARIABLES
• 1 = VARIABLES WITHOUT BAR/PRIME
• 0 = VARIABLES WITH BAR/PRIME
NOTE:
EACH MINTERM MUST CONTAIN ALL VARIABLES !!
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13. EXAMPLE
VARIABLES = X , Y
POSSIBLE
COMBINATIONS = 22
= 4 Index Bin Val Minterms MVal
0 00 x’ . y’ m0
1 01 x’ . y m1
2 10 x . y’ m2
3 11 x . y m3 5/29/2020UET 13

14. • 4 MINTERMS FOR 2 VARIABLES.
• ? MINTERMS FOR 3 VARIABLES.
• ? MINTERMS FOR 4 VARIABLES.
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15. MAXTERMS
• OBTAINED BY OR-ING ALL VARIABLES
• 1 = VARIABLES WITH BAR/PRIME
• 0 = VARIABLES WITHOUT BAR/PRIME
NOTE:
EACH MAXTERM MUST CONTAIN ALL VARIABLES !!
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16. EXAMPLE
VARIABLES = X , Y
POSSIBLE
COMBINATIONS = 22
= 4 Index Bin Val Maxterms MVal
0 00 x + y M0
1 01 x + y’ M1
2 10 x’ + y M2
3 11 x’ + y’ M35/29/2020LCWU 16

17. RELATIONSHIP BTW MINTERMS
AND MAXTERMS
MINTERMS = MI
MAXTERMS = MJ
MI = MJ
AND
MJ = MI
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