1) The document discusses digital logic concepts including number systems, logic gates, Boolean algebra, and Karnaugh maps. 2) Digital logic operates with two signal levels represented by 1 and 0. Boolean algebra describes logic circuits using variables limited to two values. 3) Karnaugh maps provide a method to simplify Boolean logic expressions by grouping adjacent 1s in the truth table.

1. PE-4030
Weeks 11 & 12
Chapter 8:
Digital Logic
Professor Charlton S. Inao
Professor , Mechatronics System Design
Defence University
College of Engineering
Bishoftu, Ethiopia

2. Instructional Objectives
To explain and understand the following
concept:
1. Digital Logic
2. Number Systems
3. Logic Gates
4. Boolean Algebra
5. Karnaugh Maps

3. Digital Logic
• Digital circuitry is the basis of digital computers
and microprocessor controlled systems.
• Digital logic operates with digital signals where
there are only two possible signal levels.
• This circuitry evolved from the transistor circuits
being able to output at one of two voltage levels
depending on the levels at its inputs. The two
levels usually 5V and 0V are the high and low
signals and represented by 1 and 0.

4. Number System In Mechatronics
Decimal
Binary
0
0000
1
BCD
Octal
Hexadecimal
00000000
0
0
0001
0000 0001
1
1
2
0010
0000 0010
2
2
3
0011
0000 0011
3
3
4
0100
0000 0100
4
4
5
0101
0000 0101
5
5
6
0110
0000 0110
6
6
7
0111
0000 0111
7
7
8
1000
0000 1000
10
8
9
1001
0000 1001
11
9
10
1010
0001 0000
12
A
11
1011
0001 0001
13
B
12
1100
0001 0010
14
C
13
1101
0001 0011
15
D
14
1110
0001 0100
16
E
15
1111
0001 0101
17
F

23. LOGIC GATES

24. AND, OR , NOT

25. NAND , NOR, EXOR

26. Boolean Algebra
• An algebraic system that describes the logic circuit,
in which the variables are limited to two values,
usually 0 and 1.
• George Boole developed an algebra for values for
the systematic treatment of logic.
• Boolean algebra deals with variables that take on
two discrete values, 0 and 1 , and with operations
that assume logical meaning.
• Situations involving “yes-no, true –false,on-off” can
be represented by Boolean Logical operations.

27. OR
Boolean Algebra Laws
1) A + 1= 1
2) A + 0 = A
3) A.0 = 0
4) A.1 = A
5) A + A =A elec
6) A.A = A elec
7)A.A = 0 elec
8) A + A = 1 elec
9) A + B = B + A elec
OR
AND
OR
AND
AND
OR
10) AB + AC= A(B + C)elec
elec
11) A + BC =(A+B)(A+C)
12) A + B = A.B elec
13) A.B = A + B
14) AΦ B= A.B + A.B
elec
15) A + AB = A + B
NAND
(exor)
elec

28. Boolean Algebra Laws
1) Anything Ored to itself is equal to itself. A + A =A
2) Anything ANDed to itself is equal to itself. A . A =A
3) It does not matter in which order we consider
inputs for OR and AND gates.
A+B=B+A
A.B=B.A
4) We can use truth table to show we can treat
bracketed terms in the same way as the ordinary
algebra.
A. (B +C)=A.B + A.C
A +(B.C) =( A+B) . (A+C)

29. Boolean Algebra Laws
5) Anything ORed with its own inverse equals 1.
A +A =1
6) Anything ANDed with its own inverse equals =0
A.A=0
7) Anything Ored with a zero is equal to itself. Anything
Ored with a 1 is equal to 1.
A + 0 =A ; A + 1= 1
8) Anything ANDed with a 0 is equal to zeo; anything
ended with 1 is equal to itself.
A.0 = 0
A.1 = A

30. Six Axioms on Properties of Boolean Algebra
Commutative Axiom:
A.B=B.A
A+B=B+A
Distributive Axiom:
A.(B+C)=(A.B) +(A.C)
A+(B.C)=(A+B ).(A+C)
Idempotency Axiom:
A.A=A
A+A=A
Absorption Axiom
A.(A +B)=A
A +(A.B)=A
Complementation Axiom
A.A=0
A+A= 1
A=A
De Morgan’s theorem
A.B= A + B
A+B= A. B

31. De Morgan’s Law

35. Application of Logic Gates
Application No.1

38. Application No: 2
• A system uses 3 switches: A,B and C. A
combination of the three switches determines
whether an alarm ,X, will make a sound.
• If switch A or B are in the ON position, and if
switch C is in the OFF position then a signal to
sound an alarm X, is produced.

39. • Solution
• 1)Construct a Truth Table for 3 inputs ,
A,B,C(23=8)
A
B
C
1
0
0
0
2
0
0
1
3
0
1
0
4
0
1
1
5
1
0
0
6
1
0
1
Output
X
0
0
1
0
1
0
1
0

40. • 2) Get the value of P and Q, form the logic of
A,B,and C based on the logic circuit.
A
B
P
C
Q=c
1
0
0
0
0
1
2
0
0
0
1
0
0
1
3
0
1
1
1
0
4
0
1
1
0
1
5
1
0
1
1
0
0
1
6
1
1
0
0
1

41. P and Q= X
P
Q
X
0
1
0
0
0
0
1
1
1
1
0
0
1
1
1
1
0
0
1
1
1
1
0
0
(A+B).C=X

42. Karnaugh Maps

47. Three Variable Map

48. Four Variable Map

49. Four Variable Map

52. Application of Karnaugh Map

53. Karnaugh Map

54. K- Map Overlapping Group

61. The End