1. Speaker: Fuw-Yi Yang 楊伏夷 伏夷非征番, 道德經 察政章(Chapter 58) 伏者潛藏也
數位系統  Digital Systems 
Department of Computer Science and Information Engineering, Chaoyang University of Technology
朝陽科技大學資工系
Speaker: Fuw-Yi Yang 楊伏夷
伏夷非征番,
道德經 察政章(Chapter 58) 伏者潛藏也
道紀章(Chapter 14) 道無形象, 視之不可見者曰夷
Fuw-Yi Yang

2. Text Book: Digital Design 5th ed.
Chap 2 Boolean Algebra and Logic Gates
2.1 Introduction
2.2 Basic Definitions
2.3 Axiomatic Definition of Boolean Algebra
2.4 Basic Theorems and Properties of Boolean Algebra
2.5 Boolean Functions
2.6 Canonical and Standard Forms
2.7 Other Logic Operations
2.8 Digital Logic Gates
2.9 Integrated Circuits
Fuw-Yi Yang

3. Text Book: Digital Design 5th ed. Chap 2.1 Introduction
Because binary logic is used in all of today’s digital computers and devices, the cost of the circuit that implement it is an important factor addressed by designers.
Finding simpler and cheaper, but equivalent, realizations of a circuit can reap huge payoffs in reducing the overall cost of the design.
Fuw-Yi Yang

4. Text Book: Digital Design 5th ed. Chap 2.2 Basic Definitions
An axiom, or postulate, is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.
The most common postulates used to formulate various algebraic structures are as follows:
1. Closure. A set S is closed with respect to a binary
operator  if, for every a, b  S, a  b  S.
2. Associative law. (a  b)  c =a  (b  c) for a, b, c  S.
3. Commutative law. a  b = b  a for a, b S.
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5. Text Book: Digital Design 5th ed. Chap 2.2 Basic Definitions
4. Identity element. If there exists an element e S such
that e  b = b  e = b for b S.
5. Inverse. For a S, if there exists an element b S such
that a  b = e, b is called the inverse of a.
6. Distributive law. If  and * are two binary operators on a
set S, * is said to be distributive over  whenever
a * (b  c) = (a * b)  (a * c).
Fuw-Yi Yang

6. Text Book: Digital Design 5th ed.
Chap 2.3 Axiomatic Definition of Boolean Algebra
In 1854, George Boole developed an algebraic system now called Boolean Algebra.
Boolean algebra is an algebraic structure defined by a set of elements, B, together with two binary operators,
+ and ∙, if the following postulates are satisfied:
1. Closure. With respect to the operators + and ∙.
2. Identity element.
The element 0 is an identity element w.r.t. +;
i.e. x + 0 = 0 + x = x.
Fuw-Yi Yang

7. Text Book: Digital Design 5th ed.
Chap 2.3 Axiomatic Definition of Boolean Algebra
The element 1 is an identity element w.r.t. ∙;
i.e. x ∙1 =1 ∙x = x.
3. Commutative law. W.r.t. the operators + and ∙;
i.e. a + b = b + a;
a ∙b = b ∙a.
4. Distributive law. a + (b ∙ c) = (a + b) ∙(a + c);
a ∙ (b + c) = (a ∙b) + (a ∙c);
5. Complement. for a B, there exists an element a' B
such that a + a' = 1, and a ∙a' = 0.
6. There exist at least two elements a, b B, such that a  b.
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8. Text Book: Digital Design 5th ed.
Chap 2.3 Axiomatic Definition of Boolean Algebra –
Example of an algebraic structure
A two-valued Boolean algebra is defined on a set of two elements, B = {0, 1}, with rules for the two binary operators + (OR) and * (AND) as shown in the following tables: x y
x*y 1 x y
x+y 1 x x' 1
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9. Text Book: Digital Design 5th ed.
Chap 2.3 Axiomatic Definition of Boolean Algebra
Show that the two-valued Boolean algebra defined above satisfies postulates 1~6.
代數結構 (群組元素, 運算操作) 具備 特性 或 性質
例如: (B = {0, 1}; *, +) Postulates 1~6
找合適的 實務的 電子電路 或 實作環境
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10. Text Book: Digital Design 5th ed.
Chap 2.4 Basic Theorems and Properties of Boolean
Algebra
Table 2.1 lists six theorems of Boolean algebra and four of its postulates.
Note that the property of Duality—
every algebraic expression deducible from the postulates of Boolean algebra remains valid if the operators and identity elements are interchanged. (part a and part b in Table 2.1)
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11. Text Book: Digital Design 4th Ed.
Chap 2.4 Basic Theorems and Properties of Boolean
Algebra
Table 2.1 lists six theorems of Boolean algebra and four of its postulates.
Fuw-Yi Yang

12. Text Book: Digital Design 5th ed.
Chap 2.4 Basic Theorems and Properties of Boolean
Algebra
Theorem 1 (a) x + x = x
Statement
Justification
x + x = (x + x) * 1
= (x + x) * (x + x')
= x + (x * x')
= x + 0
= x
Postulate 2(b), identity
Postulate 5(a), x + x' = 1 Postulate 4(b),
x + (y * z) = (x + y) * (x + z)
Postulate 5(b), x * x' = 0
Postulate 2(a), x + 0 = x
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13. Text Book: Digital Design 5th ed.
Chap 2.4 Basic Theorems and Properties of Boolean
Algebra
Theorem 1 (b) x * x = x
Statement
Justification
x * x = (x * x) + 0
= (x * x) + (x * x')
= x * (x + x')
= x * 1
= x
Postulate 2(a), identity
Postulate 5(b), x * x' = 0 Postulate 4(a),
x * (y + z) = (x * y) + (x * z)
Postulate 5(a), x + x' = 1
Postulate 2(b), x * 1= x
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14. Text Book: Digital Design 5th ed.
Chap 2.4 Basic Theorems and Properties of Boolean
Algebra
Theorem 2 (a) x + 1 = 1 (b) x * 0 = 0 by duality
Statement
Justification
x + 1 = (x + 1) * 1
= (x + 1) * (x + x')
= x + (1 * x')
= x + x'
= 1
Postulate 2(b), identity
Postulate 5(a), x + x' = 1 Postulate 4(b),
x + (y * z) = (x + y) * (x + z)
Postulate 2(b), 1 * x = x
Postulate 5(a), x + x' =1
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15. Text Book: Digital Design 5th ed.
Chap 2.4 Basic Theorems and Properties of Boolean
Algebra
Theorem 3 (x')' = x
Postulate 5:
x + x' = 1 and x * x' = 0 together define the complement of x.
The complement of x' is x and is also (x')'.
Since the complement is unique, we complete the proof.
Fuw-Yi Yang

16. Text Book: Digital Design 5th ed.
Chap 2.4 Basic Theorems and Properties of Boolean
Algebra
Theorem 6 (a) x + x * y = x (b) x * (x + y) = x by duality
Absorption
Statement
Justification
x + x * y = x * 1 + x * y
= x * (1 + y)
= x * 1
= x
Postulate 2(b), identity
Postulate 4(a),
x * (y + z) = (x * y) + (x * z)
Postulate 2(a), 1 + x = 1
Postulate 2(b), 1 * x = x
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17. Text Book: Digital Design 5th ed.
Chap 2.4 Basic Theorems and Properties of Boolean
Algebra
Theorem 4
Associative x + (y + z) = (x + y) + z
x * (y * z) = (x * y) * z
Theorem 5
DeMorgan (x + y)' = x' * y'
(x * y)' = x' + y'
Show its validity with truth table!!
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18. Text Book: Digital Design 5th ed.
Chap 2.4 Basic Theorems and Properties of Boolean
Algebra
Theorem 5 DeMorgan (x + y)' = x' * y' , (x * y)' = x' + y'
(x * y)' = {a | a  (x * y)}
= {a | ((a  x ) * (a  y))' }
= {a | (a  x ) + (a  y)}
= x' + y'
(x + y)' = {a | a  (x + y)}
= {a | ((a  x ) + (a  y))' }
= {a | (a  x ) * (a  y)}
= x' * y'
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19. Text Book: Digital Design 5th ed.
Chap 2.4 Basic Theorems and Properties of Boolean
Algebra
The Operator Precedence for evaluating Boolean expressions is:
1. parentheses
2. NOT
3. AND
4. OR
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20. Text Book: Digital Design 5th ed. Chap 2.5 Boolean Functions
Boolean algebra is an algebra that deals with binary variables and logic operations.
A Boolean function described by an algebraic expression consists of binary variables, the constant 0 and 1, and the logic operation symbols.
For a given value of the binary variables, the function can be equal to either 0 or 1.
Example: F = x + yz, F is equal to 1 if x is equal to 1 or if both y and z are equal to 1.
Fuw-Yi Yang

21. Text Book: Digital Design 5th ed. Chap 2.5 Boolean Functions
A Boolean function can be represented in a truth table. The number of rows in the truth table is 2n, where n is the number of variables in the function.
Table 2.2 shows the truth table for the function F1 = x + y'z.
Can we derive the Boolean function described by the column F2 ?
Can we draw the gate implementation of F1 and F2 ?
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22. Text Book: Digital Design 5th ed. Chap 2.5 Boolean Functions
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23. Text Book: Digital Design 4th Ed. Chap 2.5 Boolean Functions
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24. Text Book: Digital Design 5th ed. Chap 2.5 Boolean Functions
When a Boolean expression is implemented with logic gates, each term requires a gate and each variable within the term designates an input to the gate. We define a literal to be a single variable within a term, in complemented or un-complemented form.
Example 2.1 Simplify the following Boolean functions to a minimum number of literals.
1. x (x' + y) 2. x + x' y
3. (x + y) (x + y') 4. x y + x' z + y z 5. (x + y) (x' + z) (y+ z)
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25. Text Book: Digital Design 4th ed. Chap 2.5 Boolean Functions
The complement of a function F is F' and is obtained from
an interchanges of 0’s for 1’s and 1’s for 0’s in the value of
F. It can be derived algebraically through DeMorgan’s
theorem. (A + B)' = A' B' ; (A B)' = A' + B'
Example 2.2 Find the complement of the functions
F1 = x' y z' + x' y' z; F2 = x (y' z' + y z)
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26. Text Book: Digital Design 5th ed. Chap 2.5 Boolean Functions
Example 2.3 Find the complement of the functions
F1 = x' y z' + x' y' z and F2 = x (y' z' + y z) by taking their
duals and complementing each literal.
F1 = x' y z' + x' y' z
The dual of F1 is (x' + y + z')(x' + y' + z)
Complement each literal: F'1 = (x + y' + z)(x + y + z')
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27. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
A binary variable may appear either in its normal form (x)
or in its complement form (x').
For two binary variables x and y combined with an AND
Operation, we have four possible combinations:
x y, x' y, x y', x' y'.
Each of these four AND terms is called a minterm, or a
standard product.
In a similar manner, n variables can be combined to form
2n minterms.
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28. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
A binary variable may appear either in its normal form (x)
or in its complement form (x').
For two binary variables x and y combined with an OR
Operation, we have four possible combinations:
x + y, x' + y, x + y', x' + y'.
Each of these four OR terms is called a maxterm, or a
standard sum.
In a similar manner, n variables can be combined to form
2n maxerms.
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29. Text Book: Digital Design 4th Ed.
Chap 2.6 Canonical and Standard forms
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30. Text Book: Digital Design 4th Ed.
Chap 2.6 Canonical and Standard forms
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31. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
A Boolean function can be expressed algebraically from a
given truth table by forming a minterm for each
combination of the variables that produces a 1 in the
function and then taking the OR of all those terms.
See Table 2.3, 2.4
f1 = x'y'z + xy'z' + xyz =m1 + m4 + m7
f2 = x'yz + xy'z + xy'z' + xyz =m3 + m5 + m6 + m7
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32. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Now consider the complement of a Boolean function. It can
be expressed algebraically from a given truth table by
forming a minterm for each combination of the variables
that produces a 0 in the function and then taking the OR of
all those terms.
f1 = x'y'z + xy'z' + xyz =m1 + m4 + m7
f '1 = x'y'z' + x'yz' + x'yz + xy'z + xy'z' = m0+m2+m3+m5+m6
If we take the complement of f '1 we obtain the function f 1:
f 1 = (x+y+z)(x+y'+z)(x+y'+z')(x'+y+z')(x'+y+z)
=M0 M2 M3 M5 M6
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33. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Sum of minterms
Boolean functions expressed as a sum of minterms or
product of maxterms are said to be in canonical form.
Sum of minterms:
Example 2.4 Express the Boolean function F = A + B'C as a
sum of minterms.
F = A + B'C = A(B + B') (C + C') + (A + A') B'C
= ABC+ABC'+AB'C+AB'C'+AB'C+A'B'C
= ABC+ABC'+AB'C+AB'C'+A'B'C
= m7 + m6 + m5 + m4 + m1
Or F(A, B, C) = (1, 4, 5, 6, 7)
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34. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Sum of minterms
Example 2.4' Deriving the minterms of a Boolean function directly from the given truth table.
See next page
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35. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Sum of minterms
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36. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Product of maxterms
Product of maxterms:
Example 2.5 Express the Boolean function F = xy + x'z as a
product of maxterms.
F = xy + x'z
= (xy + x')(xy + z) distributive law
= (x + x')(y + x')(x + z)(y + z) distributive law
= (y + x' + zz' )(x + z + yy' )(y + z + xx' )
= (y+x'+z') (y+x'+z) (x+z+y') (x+z+y) (y+z+x') (y+z+x)
= (y+x'+z') (y+x'+z) (x+z+y')(x+z+y)
= M5M4M2M Or F(x, y, z) = (0, 2, 4, 5)
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37. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
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.
Example: F(A, B, C) = (1, 4, 5, 6, 7)
F'(A, B, C) = (0, 2, 3) = m0 + m2 + m3
Now, take the complement of F' by DeMorgan’s theorem,
we obtain F in a different form:
F(A, B, C) = (m0 + m2 + m3)' = m'0m'2m'3
= M0M2M3 = (0, 2, 3)
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38. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Conversion between Canonical Forms
A Boolean function can be expressed as product of
maxterms or sum of minterms directly from its truth table.
F(x, y, z) = (1, 3, 6, 7)
F(x, y, z) = (0, 2, 4, 5)
Table 2.6 F = xy + x'z x y z F 1
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39. Text Book: Digital Design 4th ed.
Chap 2.6 Canonical and Standard forms
Standard Forms
The two canonical forms of Boolean algebra are basic
forms that one obtains from reading a given function from
the truth table. These forms are very seldom the ones with
the least number of literals, because each minterm or
maxterm must contain, by definition, all the variables,
either complemented or un-complemented.
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40. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Standard Forms
Another way to express Boolean functions is in standard
form. In this configuration, the terms that form the function
may contain one, two, or any number of literals. There are
two type of standard forms: the sum of products (SOP) and
products of sums (POS).
Both POS and SOP are referred to as two-level
implementation.
See next page
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41. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Standard Forms—two level ckts
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42. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Standard Forms—two level ckts
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43. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Non-standard Forms
A Boolean functions may be expressed in a nonstandard
form. The implementation may requires three levels of
gating in this circuit.
See next page
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44. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Non-standard Forms
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45. Text Book: Digital Design 5th ed.
Chap 2.6 Canonical and Standard forms
Non-standard Forms
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46. Text Book: Digital Design 5th ed. Chap 2.7 Other logic operations
When the binary operators AND and OR are placed
between two variables, x and y, they form two Boolean
functions, xy and x+y, respectively. Previously, we stated
that there are 22n functions for n binary variables.
see next pages x y
f:16 combinations
0/1, two possible values 1
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47. Text Book: Digital Design 5th ed. Chap 2.7 Other logic operations
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48. Text Book: Digital Design 4th Ed. Chap 2.7 Other logic operations
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49. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
Since Boolean functions are expressed in terms of AND,
OR, and NOT operations, it is easier to implement a
Boolean function with these type of gates.
Still, the possibility of constructing gates for the other logic
operations is of practical interest.
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50. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
Factors to be weighed in considering the construction of the
other types of logic gates are:
1. the feasibility and economy of producing the gate
2. the possibility of extending the gate to more inputs
3. the basic properties of the binary operator, such as commutativity and associativity
4. the ability of the gate to implement Boolean functions alone or in conjunction with other gates
See next pages
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51. Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates
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52. Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates
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53. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
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54. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
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55. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
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56. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
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57. Text Book: Digital Design 4th Ed. Chap 2.8 Digital logic gates
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58. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
---positive and negative logic
The binary signal at the inputs and outputs of any gate has
one of two values, except during transition.
One signal value represents logic 1 and the other logic 0.
Since two signal values are assigned to two logic values,
there exist two different assignments of signal level to
logic value.
See next pages
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59. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
---positive and negative logic
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60. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
---positive and negative logic
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61. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
---positive and negative logic
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62. Text Book: Digital Design 5th ed. Chap 2.8 Digital logic gates
---positive and negative logic
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63. Text Book: Digital Design 5th ed. Chap 2.9 Integrated circuits
An integrated circuit (IC) is a silicon semiconductor crystal, called a chip, containing the electronic components for constructing digital gates.
Levels of Integration:
Small-scale integration (SSI)
Medium-scale integration (MSI)
Large-scale integration (LSI)
Very large-scale integration (VLSI)
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64. Text Book: Digital Design 5th ed. Chap 2.9 Integrated circuits
Digital logic Families:
Transistor-transistor logic (TTL)
Emitter-coupled logic (ECL)
Metal-oxide semiconductor (MOS)
Complementary Metal-oxide semiconductor (CMOS)
Computer-Aided Design
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