This can be used to learn the basics of combinational logic design. Boolean functions and digital logic gates are included.

1. Combinational Logic Design
D.R.V.L.B Thambawita
August 27, 2017
D.R.V.L.B Thambawita Combinational Logic Design

2. Table of contents
D.R.V.L.B Thambawita Combinational Logic Design

3. INTRODUCTION
In digital electronics, a circuit is a network that processes discrete
valued variables.
one or more discrete-valued input terminals
one or more discrete-valued output terminals
a functional specification describing the relationship between
inputs and outputs
a timing specification describing the delay between inputs
changing and outputs responding
D.R.V.L.B Thambawita Combinational Logic Design

4. combinational or sequential
combinational circuit
circuits outputs depend only on the current values of the inputs; in
other words, it combines the current input values to compute the
output.A combinational circuit is memoryless.
sequential circuit
circuits outputs depend on both current and previous values of the
inputs; in other words, it depends on the input sequence. A
sequential circuit has memory.
D.R.V.L.B Thambawita Combinational Logic Design

5. BOOLEAN EQUATIONS
Boolean equations deal with variables that are either TRUE or
FALSE, so they are perfect for describing digital logic.
D.R.V.L.B Thambawita Combinational Logic Design

6. Sum-of-Products Form
Do you know?
Sum-of-Products Form
This is called the sum-of-products canonical form of a function
because it is the sum (OR) of products (ANDs forming minterms).
D.R.V.L.B Thambawita Combinational Logic Design

7. Product-of-Sums Form
Do you know?
An alternative way of expressing Boolean functions is the
product-of sums canonical form. Each row of a truth table
corresponds to a maxterm that is FALSE for that row.
D.R.V.L.B Thambawita Combinational Logic Design

8. BOOLEAN ALGEBRA
Just as you use algebra to simplify mathematical equations,
you can use Boolean algebra to simplify Boolean equations.
Boolean algebra is based on a set of axioms that we assume
are correct.
Axioms are unprovable in the sense that a definition cannot be
proved.
From these axioms, we prove all the theorems of Boolean
algebra.
Do you know?
These theorems have great practical significance, because they
teach us how to simplify logic to produce smaller and less costly
circuits.
D.R.V.L.B Thambawita Combinational Logic Design

9. Axioms of Boolean algebra
Do you know?
Axioms and theorems of Boolean algebra obey the principle of
duality. If the symbols 0 and 1 and the operators • (AND) and +
(OR) are interchanged, the statement will still be correct.
D.R.V.L.B Thambawita Combinational Logic Design

10. Boolean theorems of one variable
What are the benefits?
D.R.V.L.B Thambawita Combinational Logic Design

11. Theorems of Several Variables
D.R.V.L.B Thambawita Combinational Logic Design

12. Basic Theorems
D.R.V.L.B Thambawita Combinational Logic Design

13. Basic Theorems
D.R.V.L.B Thambawita Combinational Logic Design

14. De Morgans Theorem
This is a particularly powerful tool in digital design.
According to De Morgans theorem, a NAND gate is
equivalent to an OR gate with inverted inputs.
A NOR gate is equivalent to an AND gate with inverted
inputs.
D.R.V.L.B Thambawita Combinational Logic Design

15. De Morgans Theorem
you can imagine that pushing a bubble through the gate causes it
to come out at the other side and flips the body of the gate from
AND to OR or vice versa
Pushing bubbles backward (from the output) or forward (from
the inputs) changes the body of the gate from AND to OR or
vice versa.
Pushing a bubble from the output back to the inputs puts
bubbles on all gate inputs.
Pushing bubbles on all gate inputs forward toward the output
puts a bubble on the output.
D.R.V.L.B Thambawita Combinational Logic Design

16. BOOLEAN FUNCTIONS
Boolean algebra is an algebra that deals with binary variables
and logic operations.
A Boolean function can be represented in a truth table.
Example
F1 = x + y z The function F1 is equal to 1 if x is equal to 1 or
if both y’ and z are equal to 1. F1 is equal to 0 otherwise
F2 = x y z + x yz + xy
D.R.V.L.B Thambawita Combinational Logic Design

17. BOOLEAN FUNCTIONS
Truth Tables for F1 and F2
D.R.V.L.B Thambawita Combinational Logic Design

18. BOOLEAN FUNCTIONS
Gate implementation of F1 = x + y z
D.R.V.L.B Thambawita Combinational Logic Design

19. BOOLEAN FUNCTIONS
Gate implementation of F2 = x y z + x yz + xy
D.R.V.L.B Thambawita Combinational Logic Design

20. BOOLEAN FUNCTIONS
F2 = x y z + x yz + xy = x y(y + y) + xy = x y + xy
D.R.V.L.B Thambawita Combinational Logic Design

21. BOOLEAN FUNCTIONS
1 x(x + y)
D.R.V.L.B Thambawita Combinational Logic Design

22. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy
D.R.V.L.B Thambawita Combinational Logic Design

23. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy
D.R.V.L.B Thambawita Combinational Logic Design

24. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
D.R.V.L.B Thambawita Combinational Logic Design

25. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y
D.R.V.L.B Thambawita Combinational Logic Design

26. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y = (x + x )(x + y)
D.R.V.L.B Thambawita Combinational Logic Design

27. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y = (x + x )(x + y) = 1(x + y)
D.R.V.L.B Thambawita Combinational Logic Design

28. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y = (x + x )(x + y) = 1(x + y) = x + y.
D.R.V.L.B Thambawita Combinational Logic Design

29. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y = (x + x )(x + y) = 1(x + y) = x + y.
3 (x + y)(x + y )
D.R.V.L.B Thambawita Combinational Logic Design

30. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y = (x + x )(x + y) = 1(x + y) = x + y.
3 (x + y)(x + y ) = x + xy + xy + yy
D.R.V.L.B Thambawita Combinational Logic Design

31. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y = (x + x )(x + y) = 1(x + y) = x + y.
3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y )
D.R.V.L.B Thambawita Combinational Logic Design

32. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y = (x + x )(x + y) = 1(x + y) = x + y.
3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x.
D.R.V.L.B Thambawita Combinational Logic Design

33. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y = (x + x )(x + y) = 1(x + y) = x + y.
3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x.
4
xy + x z + yz = xy + x z + yz(x + x )
= xy + x z + xyz + x yz
= xy(1 + z) + x z(1 + y)
= xy + x z.
5 (x + y)(x + z)(y + z)
D.R.V.L.B Thambawita Combinational Logic Design

34. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y = (x + x )(x + y) = 1(x + y) = x + y.
3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x.
4
xy + x z + yz = xy + x z + yz(x + x )
= xy + x z + xyz + x yz
= xy(1 + z) + x z(1 + y)
= xy + x z.
5 (x + y)(x + z)(y + z) = (x + y)(x + z)
D.R.V.L.B Thambawita Combinational Logic Design

35. BOOLEAN FUNCTIONS
1 x(x + y) = xx + xy = 0 + xy = xy.
2 x + x y = (x + x )(x + y) = 1(x + y) = x + y.
3 (x + y)(x + y ) = x + xy + xy + yy = x(1 + y + y ) = x.
4
xy + x z + yz = xy + x z + yz(x + x )
= xy + x z + xyz + x yz
= xy(1 + z) + x z(1 + y)
= xy + x z.
5 (x + y)(x + z)(y + z) = (x + y)(x + z) by duality from
function 4
D.R.V.L.B Thambawita Combinational Logic Design

36. BOOLEAN FUNCTIONS
Complement of a Function
(A + B + C) = (A + x) let B + C = x
= A x by DeMorgan
= A (B + C) substitute B + C = x
= A (B C ) by DeMorgan
= ABC by associative
D.R.V.L.B Thambawita Combinational Logic Design

37. BOOLEAN FUNCTIONS
Complement of a Function
(A + B + C + D + ... + F) = A B C D ...F
(ABCD...F) = A + B + C + D + ... + F
D.R.V.L.B Thambawita Combinational Logic Design

38. BOOLEAN FUNCTIONS
Find the complement of the functionsF1 = x yz + x y z and
F2 = x(y z + yz)
D.R.V.L.B Thambawita Combinational Logic Design

39. BOOLEAN FUNCTIONS
Find the complement of the functionsF1 = x yz + x y z and
F2 = x(y z + yz)
D.R.V.L.B Thambawita Combinational Logic Design

40. BOOLEAN FUNCTIONS
Do you know?
D.R.V.L.B Thambawita Combinational Logic Design

41. More about Minterms and Maxterms
Minterms
Consider two binary variables x and y combined with an AND
operation
D.R.V.L.B Thambawita Combinational Logic Design

42. More about Minterms and Maxterms
Minterms
Consider two binary variables x and y combined with an AND
operation
There are four possible combinations: x’y’, x’y, xy’, and xy
D.R.V.L.B Thambawita Combinational Logic Design

43. More about Minterms and Maxterms
Minterms
Consider two binary variables x and y combined with an AND
operation
There are four possible combinations: x’y’, x’y, xy’, and xy
Each of these four AND terms is called a minterm, or a
standard product.
D.R.V.L.B Thambawita Combinational Logic Design

44. More about Minterms and Maxterms
Minterms
Consider two binary variables x and y combined with an AND
operation
There are four possible combinations: x’y’, x’y, xy’, and xy
Each of these four AND terms is called a minterm, or a
standard product.
n variables can be combined to form 2n minterms
D.R.V.L.B Thambawita Combinational Logic Design

45. More about Minterms and Maxterms
Minterms
Consider two binary variables x and y combined with an AND
operation
There are four possible combinations: x’y’, x’y, xy’, and xy
Each of these four AND terms is called a minterm, or a
standard product.
n variables can be combined to form 2n minterms
D.R.V.L.B Thambawita Combinational Logic Design

46. Minterms and Maxterms
Maxterms
N variables forming an OR term, with each variable being
primed or unprimed, provide 2n possible combinations, called
maxterms, or standard sums.
D.R.V.L.B Thambawita Combinational Logic Design

47. Minterms and Maxterms
Maxterms
N variables forming an OR term, with each variable being
primed or unprimed, provide 2n possible combinations, called
maxterms, or standard sums.
The eight maxterms for three variables.
D.R.V.L.B Thambawita Combinational Logic Design

48. Minterms and Maxterms
Maxterms
N variables forming an OR term, with each variable being
primed or unprimed, provide 2n possible combinations, called
maxterms, or standard sums.
The eight maxterms for three variables.
D.R.V.L.B Thambawita Combinational Logic Design

49. Minterms and Maxterms
Minterms and Maxterms for Three Binary Variables
D.R.V.L.B Thambawita Combinational Logic Design

50. Minterms and Maxterms
Example: Functions of Three Variables
D.R.V.L.B Thambawita Combinational Logic Design

51. Minterms and Maxterms
Fuction f1 and f2 for above truth table
D.R.V.L.B Thambawita Combinational Logic Design

52. Minterms and Maxterms
Fuction f1 and f2 for above truth table
f1 = x y z + xy z + xyz = m1 + m4 + m7
D.R.V.L.B Thambawita Combinational Logic Design

53. Minterms and Maxterms
Fuction f1 and f2 for above truth table
f1 = x y z + xy z + xyz = m1 + m4 + m7
f2 = x yz + xy z + xyz + xyz = m3 + m5 + m6 + m7
D.R.V.L.B Thambawita Combinational Logic Design

54. Minterms and Maxterms
Fuction f1 and f2 for above truth table
f1 = x y z + xy z + xyz = m1 + m4 + m7
f2 = x yz + xy z + xyz + xyz = m3 + m5 + m6 + m7
D.R.V.L.B Thambawita Combinational Logic Design

55. PRODUCT-OF-SUMS
Following table shows the truth table for a Boolean function, Y,
and its complement, ¯Y . Using De Morgans Theorem, derive the
product-of-sums canonical form of Y from the sum-of-products
form of ¯Y .
D.R.V.L.B Thambawita Combinational Logic Design

56. PRODUCT-OF-SUMS
Solution:
¯Y = ¯A ¯B + ¯AB (1)
¯¯y = y = ¯A ¯B + ¯AB = ( ¯A ¯B)( ¯AB) = (A + B)(A + ¯B) (2)
D.R.V.L.B Thambawita Combinational Logic Design

57. PRODUCT-OF-SUMS
To express a Boolean function as a product of maxterms, it
must first be brought into a form of OR terms.
This may be done by using the distributive law,
x + yz = (x + y)(x + z).
Example: Express the Boolean function F = xy + xz as a product
of maxterms
D.R.V.L.B Thambawita Combinational Logic Design

58. PRODUCT-OF-SUMS
To express a Boolean function as a product of maxterms, it
must first be brought into a form of OR terms.
This may be done by using the distributive law,
x + yz = (x + y)(x + z).
Example: Express the Boolean function F = xy + xz as a product
of maxterms
F = xy + x y = (xy + x )(xy + z)
= (x + x )(y + x )(x + z)(y + z)
= (x + y)(x + z)(y + z)
D.R.V.L.B Thambawita Combinational Logic Design
Z

59. Product of Maxterms
The function has three variables: x, y, and z. Each OR term is
missing one variable; therefore
D.R.V.L.B Thambawita Combinational Logic Design
F = (x + y + z)(x + y + z)(x + y + z)(x + y + z)
= M0M2M4M5

60. Conversion between Canonical Forms
The complement of a function expressed as the sum of
minterms equals the sum of minterms missing from the
original function
Now, if we take the complement of F’ by DeMorgan’s
theorem, we obtain F in a different form:
D.R.V.L.B Thambawita Combinational Logic Design

61. Conversion between Canonical Forms
D.R.V.L.B Thambawita Combinational Logic Design

62. Standard Forms
Another way to express Boolean functions is in standard form
The sum of products is a Boolean expression containing AND
terms, called product terms, with one or more literals each
A product of sums is a Boolean expression containing OR
terms, called sum terms.
D.R.V.L.B Thambawita Combinational Logic Design

63. Standard Forms
D.R.V.L.B Thambawita Combinational Logic Design

64. Standard Forms: Three and twolevel implementation
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65. OTHER LOGIC OPERATIONS
There are 22n functions for n binary variables. Thus, for two
variables, n=2, and the number of possible Boolean functions is 16.
Truth Tables for the 16 Functions of Two Binary Variables
D.R.V.L.B Thambawita Combinational Logic Design

66. OTHER LOGIC OPERATIONS
Boolean Expressions for the 16 Functions of Two Variables
D.R.V.L.B Thambawita Combinational Logic Design

67. OTHER LOGIC OPERATIONS
Boolean Expressions for the 16 Functions of Two Variables
D.R.V.L.B Thambawita Combinational Logic Design

68. DIGITAL LOGIC GATES
D.R.V.L.B Thambawita Combinational Logic Design

69. DIGITAL LOGIC GATES
D.R.V.L.B Thambawita Combinational Logic Design

70. DIGITAL LOGIC GATES
D.R.V.L.B Thambawita Combinational Logic Design

71. DIGITAL LOGIC GATES
D.R.V.L.B Thambawita Combinational Logic Design

72. Extension to Multiple Inputs
The gates shown can be extended to have more than two
inputs.
A gate can be extended to have multiple inputs if the binary
operation it represents is commutative and associative.
Ex: OR function
x + y = y + x(commutative)
(x + y) + z = x + (y + z) = x + y + z(associative)
D.R.V.L.B Thambawita Combinational Logic Design

73. Extension to Multiple Inputs
NAND and NOR functions are commutative
NAND and NOR operators are not associative
D.R.V.L.B Thambawita Combinational Logic Design
Z

74. Extension to Multiple Inputs
Threeinput exclusiveOR function
Figure: Threeinput exclusive-OR gate
Do you know?
F is equal to 1 if only one input is equal to 1 or if all three inputs
are equal to 1. (when the total number of 1’s in the input variables
is odd)
D.R.V.L.B Thambawita Combinational Logic Design

75. Positive and Negative Logic
Do you know?
Choosing the highlevel H to represent logic 1 defines a positive
logic system. Choosing the lowlevel L to represent logic 1
defines a negative logic system.
Figure: Signal assignment and logic polarity
D.R.V.L.B Thambawita Combinational Logic Design

76. Positive and Negative Logic
D.R.V.L.B Thambawita Combinational Logic Design

77. Positive and Negative Logic
D.R.V.L.B Thambawita Combinational Logic Design

78. Positive and Negative Logic
D.R.V.L.B Thambawita Combinational Logic Design