Number System conversions


 Shona Hutchinson
 1 years ago
 Views:
Transcription
1 Number System conversions
2 Number Systems The system used to count discrete units is called number system. There are four systems of arithmetic which are often used in digital electronics. Decimal Number System Binary Number System Octal Number System Hexa Decimal System
3 Decimal Number system Decimal number system contains 0 digits: 0,,2,3,4,5,6,7,8,9; and that is why its base or radix is 0. Here radix means total number of digits used in any system. To indicate digits greater than 9 (nine) we shall use the sum of these digits times power of 0.
4 Fractional DECIMAL SYSTEM Decimal System The decimal system is composed of 0 numerals or symbols. These 0 symbols are 0,, 2, 3, 4, 5, 6, 7, 8, 9; using these symbols as digits of a number, we can express any quantity. The decimal system, also called the base 0 system because it has 0 digits
5 Decimal Number System The decimal number system is a positional number system. Example: X 0 0 = X 0 = 20 6 X 0 2 = X 0 3 = 5000
6 Binary Number System The binary number system is also a positional numbering system. Instead of using ten digits, 09, the binary system uses only two digits, 0 and.
7 Binary Number System The binary number system is also known as base 2. The values of the positions are calculated by taking 2 to some power. Why is the base 2 for binary numbers? o Because we use 2 digits, the digits 0 and.
8 Fractional BINARY SYSTEM In the binary system, there are only two symbols or possible digit values, 0 and. This base2 system can be used to represent any quantity that can be represented in decimal or other number system In digital systems the information that is being processed is usually presented in binary form. Binary quantities can be represented by any device that has only two operating states or possible conditions. Eg. a switch has only open or closed. We arbitrarily (as we define them) let an open switch represent binary 0 and a closed switch represent binary. Thus we can represent any binary number by using series of switches.
9 Octal Number System Also known as the Base 8 System Uses digits 07 Readily converts to binary Groups of three (binary) digits can be used to represent each octal digit Also uses multiplication and division algorithms for conversion to and from base 0
10 Fractional OCTAL NUMBER SYSTEM The octal number system has a base of eight, meaning that it has eight possible digits: 0,,2,3,4,5,6, Use to represent long binary numbers in computers and microprocessors.
11 Hexadecimal Number Base 6 system Uses digits 09 & letters A,B,C,D,E,F Groups of four bits represent each base 6 digit System
12 Fractional HEXADECIMAL NUMBER SYSTEM The hexadecimal system uses base 6. Thus, it has 6 possible digit symbols. It uses the digits 0 through 9 plus the letters A, B, C, D, E, and F as the 6 digit symbols Use to represent long binary numbers in computers and microprocessors. These digits can use to program machine language.
13 Binary Numbering Scale Base 2 Number Base 0 Equivalent Power Positional Value
14 Significant Digits Binary: 00 Most significant digit Least significant digit Hexadecimal: D63A7A Most significant digit Least significant digit
15 Converting From Decimal to Binary Make a list of the binary place values up to the number being converted. Perform successive divisions by 2, placing the remainder of 0 or in each of the positions from right to left. Continue until the quotient is zero. Example:
16 DECIMAL TO BINARY CONVERTION There are two methods to convert it: i. Revese of BinaryToDigital Method Decimal number write as the sum of square 0 & is write on the byte Example : Convert 450 to the binary value Solve = 45 = = = 0 0 2
17 ii. Repeat division method The numbers is divide by 2. Balance for the question is written until the last answer. Example : convert 250 to binary Solve = 250 =?2 = balance LSB = balance 0 = balance 0 = 3 balance 2 = 2 0 balance MSB... Answer = 002
18 Decimal to Binary Conversion The easiest way to convert a decimal number to its binary equivalent is to use the Division Algorithm This method repeatedly divides a decimal number by 2 and records the quotient and remainder The remainder digits (a sequence of zeros and ones) form the binary equivalent in least significant to most significant digit sequence
19 Division Algorithm Convert 67 to its binary equivalent: 67 0 = x 2 Step : 67 / 2 = 33 R Step 2: 33 / 2 = 6 R Step 3: 6 / 2 = 8 R 0 Step 4: 8 / 2 = 4 R 0 Step 5: 4 / 2 = 2 R 0 Step 6: 2 / 2 = R 0 Divide 67 by 2. Record quotient in next row Again divide by 2; record quotient in next row Repeat again Repeat again Repeat again Repeat again Step 7: / 2 = 0 R STOP when quotient equals
20 Decimal to binary Examples: (fractional) Convert the decimal number (2.0625) 0 into binary number. Solution: Fractional part: x 2 = x 2 = x 2 = x 2 =.000 (2.0625) 0 = (00.000) 2
21 Decimal to Octal Examples: Conversion (35) 0 = (473) LSD MSD
22 Convert from decimal to octal by using the repeated division method used for decimal to binary conversion. Divide the decimal number by 8 The first remainder is the LSB and the last is the MSB. Example : convert 3590 to Decimal Value Solve = DECIMAL TO OCTAL CONVERTION 3590 =?8 = balance 7 LSB 44 = 8 5 balance 4 = balance 5... Answer = 5478 MSB
23 Decimal to Octal Conversion Convert to its octal equivalent: 427 / 8 = 53 R3 Divide by 8; R is LSD 53 / 8 = 6 R5 Divide Q by 8; R is next digit 6 / 8 = 0 R6 Repeat until Q =
24 Decimal to Octal Examples: (fractional) Convert the decimal number ( ) 0 into octal number. Solution: Fractional part: x 8 = x 8 = x 8 = x 8 = x 8 = ( ) 0 = (34.634) 8
25 Decimal to Hexadecimal Conversion Examples (35) 0 = (3B) B 3 LSD MSD
26 Decimal to Hexadecimal Conversion Convert to its hexadecimal equivalent: 830 / 6 = 5 R4 5 / 6 = 3 R3 3 / 6 = 0 R3 = E in Hex 33E 6
27 Decimal to Hexadecimal (fractional) Examples: Convert the decimal number ( ) 0 into hexadecimal number. Solution: Fractional part: x 6 = x 6 = ( ) 0 = (E.39) 6
28 Converting from Binary to Decimal Example of a binary number and the values of the positions:
29 Converting from Binary to Decimal X 2 0 = X 2 = = 2 4 = 6 X 2 3 = 8 2 = = 32 0 X 2 4 = 0 X 2 2 = = = 64 0 X 2 5 = = 8 X 2 6 =
30 Binary to Decimal Conversion The easiest method for converting a binary number to its decimal equivalent is to use the Multiplication Algorithm Multiply the binary digits by increasing powers of two, starting from the right Then, to find the decimal number equivalent, sum those products
31 Multiplication Algorithm Convert (000) 2 to its decimal equivalent: Binary Positional Values x x x x x x x x Products
32 Converting from Binary to Decimal Practice conversions: Binary Decimal
33 Converting From Decimal to Binary Practice conversions: Decimal Binary
34 BINARY TO OCTAL CONVERSION Can be converted by grouping the binary bit in group of three starting from LSB Octal is a base8 system and equal to two the power of three, so a digit in Octal is equal to three digit in binary system.
35
36 binary to hexadecimal system Group the digits of the binary number by four starting from the right. 2 Replace each group of four digits by an equivalent hexadecimal digit. Convert into a hexadecimal number = B5 6 B 5
37 Exercise Convert into a hexadecimal number. 2 Convert 0 2 into a hexadecimal number.
38 Octal to Decimal Conversion Convert to its decimal equivalent: Octal Digits Positional Values Products x x x
39 OCTAL TO BINARY CONVERTION Convert from octal to binary by converting each octal digit to a three bit binary equivalent Octal digit Binary Equivalent Convert from binary to octal by grouping bits in threes starting with the LSB. Each group is then converted to the octal equivalent Leading zeros can be added to the left of the MSB to fill out the last group.
40
41 Octal to Binary Conversion Each octal number converts to 3 binary digits To convert to binary, just substitute code:
42 Hexadecimal to Decimal Conversion Convert 3B4F6 to its decimal equivalent: Hex Digits Positional Values Products 3 B 4 F x x x x ,83 0
43 HEXADECIMAL TO binary To convert a hexadecimal to binary number, convert each hexadecimal digit to its 4 bit equivalent using the hexa number. Example: (23.AB) 6 = () 2 Solution: (23.AB) 6 = 2 3. A B (23.AB) 6 = ( ) 2
44 HEXADECIMAL TO OCTAL CONVERTION There is two ways to convert it: i. Hexadecimal Decimal Octal ii. Hexadecimal Binary Octal i. Hexadecimal Decimal Octal
45 i. Hexadecimal Binary Octal
46 Hexadecimal Number System Binary Decimal Hexadecimal Binary Decimal Hexadecimal A 0 B C D E F
47 Binary Logic and Gates Binary variables take on one of two values. Logical operators operate on binary values and binary variables. Basic logical operators are the logic functions AND, OR and NOT. Logic gates implement logic functions. Boolean Algebra: a useful mathematical system for specifying and transforming logic functions. We study Boolean algebra as a foundation for designing and analyzing digital Boolean Algebra and Logic Gates 47
48 Binary Variables Recall that the two binary values have different names: True/False On/Off Yes/No /0 We use and 0 to denote the two values. Variable identifier examples: A, B, y, z, or X for now RESET, START_IT, or ADD later Boolean Algebra and Logic Gates 48
49 Logical Operations The three basic logical operations are: AND OR NOT AND is denoted by a dot ( ). OR is denoted by a plus (+). NOT is denoted by an overbar ( ), a single quote mark (') after, or (~) before the variable. Boolean Algebra and Logic Gates 49
50 Examples: Notation Examples Y A B is read Y is equal to A AND B. z x y X A is read z is equal to x OR y. is read X is equal to NOT A. Note: The statement: + = 2 (read one plus one equals two ) is not the same as + = (read or equals ). Boolean Algebra and Logic Gates 50
51 Operator Definitions Operations are defined on the values "0" and "" for each operator: AND OR 0 0 = 0 0 = 0 0 = 0 = = = + 0 = + = NOT 0 0 Boolean Algebra and Logic Gates 5
52 Boolean Algebra and Logic Gates 52 Truth Tables Tabular listing of the values of a function for all possible combinations of values on its arguments Example: Truth tables for the basic logic operations: AND OR NOT X 0 0 Y Z = X Y X Y Z = X+Y X 0 Z 0 X
53 Boolean Algebra and Logic Gates 53 Truth Tables Cont d Used to evaluate any logic function Consider F(X, Y, Z) = X Y + Y Z X Y Z X Y Y Y Z F = X Y + Y Z
54 Using Switches Inputs: Boolean Algebra and Logic Gates 54 logic is switch closed logic 0 is switch open Outputs: logic is light on logic 0 is light off. NOT input: Logic Function Implementation Normallyclosed switch => NOT logic is switch open C logic 0 is switch closed Switches in parallel => OR Switches in series => AND
55 Logic Function Implementation cont d Example: Logic Using Switches A B C Light is on (L = ) for L(A, B, C, D) = A (B C + D) = A B C + A D and off (L = 0), otherwise. Useful model for relay and CMOS gate circuits, the foundation of current digital logic circuits Boolean Algebra and Logic Gates 55 D
56 Boolean Algebra and Logic Gates 56 Logic Gates In the earliest computers, switches were opened and closed by magnetic fields produced by energizing coils in relays. The switches in turn opened and closed the current paths. Later, vacuum tubes that open and close current paths electronically replaced relays. Today, transistors are used as electronic switches that open and close current
57 X Y Logic Gate Symbols and Behavior Logic gates have special symbols: AND gat e Z = X Y X Y OR gat e Z = X + Y And waveform behavior in time as follows: X NOT gat e or i nver t er Z = X X 0 0 Y 0 0 ( AND) X Y ( OR) X + Y 0 Boolean Algebra and Logic Gates 57 ( NOT) X 0 0
58 Logic Diagrams and Expressions X Y Z Boolean equations, truth tables and logic diagrams describe the same function! Truth tables are unique, but expressions and logic diagrams are not. This gives flexibility in implementing functions. Boolean Algebra and Logic Gates 58 Truth Table F X Y Z X Y Z Logic Equation F X Y Z Logic Diagram F
59 Boolean Algebra and Logic Gates 59 Gate Delay In actual physical gates, if an input changes that causes the output to change, the output change does not occur instantaneously. The delay between an input change and the output change is the gate delay denoted by t G : Input Output 0 0 tg tg tg = 0.3 ns Time (ns)
60 Boolean Algebra Invented by George Boole in 854 An algebraic structure defined by a set B = {0, }, together with two binary operators (+ and ) and a unary operator ( ). X + 0 = X 2. X. = X Identity element 3. X + = 4. X. 0 0 = 5. X + X = X 6. X. X = X Idempotence 7. X + X = 8. X. X = 0 Complement 9. X = X Involution 0. X + Y = Y + X. XY = YX Commutative 2. (X + Y) + Z = X + (Y + Z) 3. (XY) Z X(Y Z) = Associative 4. X(Y + Z) = XY XZ + 5. X + YZ = (X + Y) (X + Z) Distributive 6. X + Y = X. Y 7. X. Y = X + Y DeMorgan s Boolean Algebra and Logic Gates 60
61 Some Properties of Boolean Algebra Boolean Algebra is defined in general by a set B that can have more than two values A twovalued Boolean algebra is also know as Switching Algebra. The Boolean set B is restricted to 0 and. Switching circuits can be represented by this algebra. The dual of an algebraic expression is obtained by interchanging + and and interchanging 0 s and s. The identities appear in dual pairs. When there is only one identity on a line the identity is selfdual, i. e., the dual expression = the original expression. Sometimes, the dot symbol (AND operator) is not written when the meaning is clear Boolean Algebra and Logic Gates 6
62 Dual of a Boolean Expression Example: F = (A + C) B + 0 Boolean Algebra and Logic Gates 62 dual F = (A C + B) = A C + B Example: G = X Y + (W + Z) dual G = (X+Y) (W Z) = (X+Y) (W+Z) Example: H = A B + A C + B C dual H = (A+B) (A+C) Unless it happens (B+C) to be selfdual, the dual of an expression does not equal the expression itself H is self Are any of these functions selfdual? (A+B)(A+C)(B+C)=(A+BC)(B+C)=AB+AC+BC
63 Boolean Algebra and Logic Gates 63 Boolean Operator Precedence The order of evaluation is:. Parentheses 2. NOT 3. AND 4. OR Consequence: Parentheses appear around OR expressions Example: F = A(B + C)(C + D)
64 Boolean Algebraic Proof Example A + A B = A (Absorption Theorem) Proof Steps Justification A + A B = A + A B Identity element: A = A = A ( + B) Distributive = A + B = = A Identity element Our primary reason for doing proofs is to learn: Careful and efficient use of the identities and theorems of Boolean algebra, and How to choose the appropriate identity or theorem to apply to make forward progress, irrespective of the Boolean Algebra and Logic Gates 64 application.
65 Boolean Algebraic Proof Example 2 AB + AC + BC = AB + AC (Consensus Theorem) Proof Steps Justification = AB + AC + BC = AB + AC + BC Identity element = AB + AC + (A + A) BC Complement = AB + AC + ABC + ABC Distributive = AB + ABC + AC + ACB Commutative = AB + ABC + AC + ACB Identity element = AB (+C) + AC ( + B) Distributive = AB. + AC. +X = = AB + AC Identity element Boolean Algebra and Logic Gates 65
66 Useful Theorems Minimization X Y + X Y = Y Absorption X + X Y = X Simplification X + X Y = X + Y DeMorgan s X + Y = X Y Boolean Algebra and Logic Gates 66 Minimization (dual) (X+Y)(X+Y) = Y Absorption (dual) X (X + Y) = X Simplification (dual) X (X + Y) = X Y DeMorgan s (dual) X Y = X + Y
67 Truth Table to Verify DeMorgan s X + Y = X Y X Y = X + Y X Y X Y X+Y X Y X+Y X Y X Y X+Y Generalized DeMorgan s Theorem: X + X X n = X X 2 X n X X 2 X n = X + X X n Boolean Algebra and Logic Gates 67
68 Complementing Functions Use DeMorgan's Theorem:. Interchange AND and OR operators 2. Complement each constant and literal Example: Complement F = F = (x + y + z)(x + y + z) Example: Complement G = (a + bc)d + e Boolean Algebra and Logic Gates 68 x y z x y z
69 Expression Simplification An application of Boolean algebra Simplify to contain the smallest number of literals (variables that may or may not be complemented) A B = AB + ABCD + A C D + A C D + A B D = AB + AB(CD) + A C (D + D) + A B D = AB + A C + A B D = B(A + AD) +AC = B (A + D) + A C (has only 5 literals) Boolean Algebra and Logic Gates 69 A C D A B D A C D A B C D
70 Next Canonical Forms Minterms and Maxterms SumofMinterm (SOM) Canonical Form ProductofMaxterm (POM) Canonical Form Representation of Complements of Functions Conversions between Representations Boolean Algebra and Logic Gates 70
71 Boolean Algebra and Logic Gates 7 Minterms Minterms are AND terms with every variable present in either true or complemented form. Given that each binary variable may x appear normal (e.g., x) or complemented (e.g., ), there are 2 n minterms for n variables. Example: Two variables (X and Y) produce 2 x 2 = 4 combinations: XY X Y X Y X Y (both normal) (X normal, Y complemented) (X complemented, Y normal) (both complemented) Thus there are four minterms of two
72 Maxterms Maxterms are OR terms with every variable in true or complemented form. Given that each binary variable may appear normal (e.g., x) or complemented (e.g., x), there are 2 n maxterms for n variables. Example: Two variables (X and Y) produce 2 x 2 = 4 combinations: X Y (both normal) X Y (x normal, y complemented) X Y (x complemented, y normal) X Boolean Algebra and Logic Gates 72 Y (both complemented)
73 Two variable minterms and maxterms. 3 m 3 = x y M 3 = x + y The minterm m i should evaluate to for each combination of x and y. The maxterm is the complement of the minterm Boolean Algebra and Logic Gates 73 Minterms & Maxterms for 2 variables x y Index Minterm Maxterm m 0 = x y M 0 = x + y 0 m = x y M = x + y 0 2 m 2 = x y M 2 = x + y
74 x Boolean Algebra and Logic Gates 74 Minterms & Maxterms for 3 y z Index variables Minterm m0 = x y z m = x y z m2 = x y z m3 = x y z m4 = x y z m5 = x y z m6 = x y z m7 = x y z Maxterm M0 = x + y + z M = x + y + z M2 = x + y + z M3 = x + y + z M4 = x + y + z M5 = x + y + z M6 = x + y + z M7 = x + y + z Maxterm Mi is the complement of minterm mi
75 Purpose of the Index Minterms and Maxterms are designated with an index The index number corresponds to a binary pattern The index for the minterm or maxterm, expressed as a binary number, is used to determine whether the variable is shown in the true or complemented form For Minterms: means the variable is Not Complemented and 0 means the variable is Complemented. For Maxterms: 0 means the variable is Not Complemented and Boolean Algebra and Logic Gates 75 means the variable is Complemented.
76 Standard Order All variables should be present in a minterm or maxterm and should be listed in the same order (usually alphabetically) Example: For variables a, b, c: Maxterms (a + b + c), (a + b + c) are in standard order However, (b + a + c) is NOT in standard order Boolean Algebra and Logic Gates 76 (a + c) does NOT contain all variables Minterms (a b c) and (a b c) are in standard order However, (b a c) is not in standard order
77 SumOfMinterm (SOM) SumOfMinterm (SOM) canonical form: Sum of minterms of entries that evaluate to x y z F Minterm m = x y z m 6 = x y z m 7 = x y z F = m + m6 + m7 = (, 6, 7) = x y z + x y z + x y z Boolean Algebra and Logic Gates 77 Focus on the entries
78 SumOfMinterm Examples F(a, b, c, d) = (2, 3, 6, 0, ) F(a, b, c, d) = m 2 + m 3 + m 6 + m 0 + m a b c d + a b c d + a b c d + a b c d + a b c d G(a, b, c, d) = (0,, 2, 5) G(a, b, c, d) = m 0 + m + m a b 2 + m a b c d + a b c d + a b c d c d 5 Boolean Algebra and Logic Gates 78
79 ProductOfMaxterm (POM) ProductOfMaxterm (POM) canonical form: Product of maxterms of entries that evaluate to 0 x y z F Maxterm M 2 = (x + y + z) M 4 = (x + y + z) M 6 = (x + y + z) Focus on the 0 entries F = M2 M4 M6 = (2, 4, 6) = (x+y+z) (x+y+z) (x+y+z) Boolean Algebra and Logic Gates 79
80 F(a, b, c, d) = (, 3, 6, ) F(a, b, c, d) = M M 3 M 6 M (a+b+c+d) (a+b+c+d (a+b+c+d (a+b+c+d G(a, b, c, d) = (0, 4, 2, 5) ) ) ) G(a, b, c, d) = M 0 M 4 M 2 M 5 Boolean Algebra and Logic Gates 80 ProductOfMaxterm Examples (a+b+c+d) (a+b+c+d ) (a+b+c+d ) (a+b+c+d )
81 Boolean Algebra and Logic Gates 8 Observations We can implement any function by "ORing" the minterms corresponding to the entries in the function table. A minterm evaluates to for its corresponding entry. We can implement any function by "ANDing" the maxterms corresponding to 0 entries in the function table. A maxterm evaluates to 0 for its corresponding entry. The same Boolean function can be expressed in two canonical ways: SumofMinterms (SOM) and ProductofMaxterms (POM). If a Boolean function has fewer entries then the SOM canonical form will contain fewer literals than POM. However, if it has fewer 0 entries then the POM
82 Converting to SumofMinterms Boolean Algebra and Logic Gates 82 Form A function that is not in the SumofMinterms form can be converted to that form by means of a truth table Consider F = y + x z x y z F Minterm m 0 = x y z 0 0 m = x y z 0 0 m 2 = x y z m 4 = x y z 0 m 5 = x y z F = (0,, 2, 4, 5) = m0 + m + m2 + m4 + m5 = x y z + x y z + x y z + x y z + x y z
83 Converting to ProductofMaxterms Boolean Algebra and Logic Gates 83 Form A function that is not in the ProductofMinterms form can be converted to that form by means of a truth table Consider x y z again: F Minterm F = y + x z F = (3, 6, 7) = M 3 = (x+y+z) M 6 = (x+y+z) 0 M 7 = (x+y+z) M3 M6 M7 = (x+y+z) (x+y+z) (x+y+z)
84 Conversions Between Canonical Forms x y z F Minterm Maxterm M 0 = (x + y + z) 0 0 m = x y z 0 0 m 2 = x y z 0 m 3 = x y z M 4 = (x + y + z) 0 m 5 = x y z 0 0 M 6 = (x + y + z) m 7 = x y z F = m +m 2 +m 3 +m 5 +m 7 = (, 2, 3, 5, 7) = x y z + x y z + x y z + x y z + x y z F = M 0 M 4 M 6 = (0, 4, 6) = (x+y+z)(x+y+z)(x+y+z) Boolean Algebra and Logic Gates 84
85 Algebraic Conversion to Sumof Minterms Expand all terms first to explicitly list all minterms AND any term missing a variable v with (v + v) Example : f = x + x y f = x (y + y) + x y f = x y + x y + x y f = m 3 + m 2 + m 0 = (0, 2, 3) Boolean Algebra and Logic Gates 85 (2 variables) Example 2: g = a + b c (3 variables) g = a (b + b)(c + c) + (a + a) b c g = a b c + a b c + a b c + a b c + a b c + a b c g = a b c + a b c + a b c + a b c + a b c g = m + m 4 + m 5 + m 6 + m 7 = (, 4, 5, 6, 7)
86 Algebraic Conversion to Productof Maxterms Expand all terms first to explicitly list all maxterms OR any term missing a variable v with v v Example : f = x + x y (2 variables) Apply 2 nd distributive law: f = (x + x) (x + y) = (x + y) = (x + y) = M Example 2: g = a c + b c + a b (3 variables) g = (a c + b c + a) (a c + b c + b) (distributive) g = (c + b c + a) (a c + c + b) (x + x y = x + y) Boolean Algebra and Logic Gates 86
87 Function Complements The complement of a function expressed as a sum of minterms is constructed by selecting the minterms missing in the sumofminterms canonical form Alternatively, the complement of a function expressed by a Sum of Minterms form is simply the Product of Maxterms with the same indices Example: Given F(x, y, z) = (, 3, 5, 7) F(x, y, z) = (0, 2, 4, 6) F(x, y, z) = (, 3, 5, 7) Boolean Algebra and Logic Gates 87
88 Boolean Algebra and Logic Gates 88 Summary of Minterms and Maxterms There are 2 n minterms and maxterms for Boolean functions with n variables. Minterms and maxterms are indexed from 0 to 2 n Any Boolean function can be expressed as a logical sum of minterms and as a logical product of maxterms The complement of a function contains those minterms not included in the original function The complement of a sumofminterms is a productofmaxterms with the same indices
89 Standard Forms Standard SumofProducts (SOP) form: equations are written as an OR of AND terms Standard ProductofSums (POS) form: equations are written as an AND of OR terms Examples: A B C A B C B SOP: (A B) (A B C ) C POS: These (A B mixed C) (A forms C) are neither SOP nor POS A B C A C (A B) Boolean Algebra and Logic Gates 89
90 Standard SumofProducts Boolean Algebra and Logic Gates 90 (SOP) A sum of minterms form for n variables can be written down directly from a truth table. Implementation of this form is a twolevel network of gates such that: The first level consists of ninput AND gates The second level is a single OR gate This form often can be simplified so that the corresponding circuit is simpler.
91 Standard SumofProducts A Simplification Example: F ( A, B, C ) Writing the minterm expression: F = A B C + A B C + A B C + ABC + ABC Simplifying: F = A B C + A (B C + B C + B C + B C) F = A B C + A (B (C + C) + B (C + C)) F = A B C + A (B + B) F = A B C + A F = B C + A Simplified F contains 3 literals compared to 5 Boolean Algebra and Logic Gates 9 (SOP) (, 4, 5, 6, 7 )
92 Boolean Algebra and Logic Gates 92 AND/OR TwoLevel Implementation The two implementations for F are shown below A B C A B C A B C A B C A B C F A B C It is quite apparent which is simpler! F
93 SOP and POS Observations The previous examples show that: Canonical Forms (Sumofminterms, Productof Maxterms), or other standard forms (SOP, POS) differ in complexity Boolean algebra can be used to manipulate equations into simpler forms Simpler equations lead to simpler implementations Questions: How can we attain a simplest expression? Is there only one minimum cost circuit? The next part will deal with these issues Boolean Algebra and Logic Gates 93
94 Terms of Use All (or portions) of this material 2008 by Pearson Education, Inc. Permission is given to incorporate this material or adaptations thereof into classroom presentations and handouts to instructors in courses adopting the latest edition of Logic and Computer Design Fundamentals as the course textbook. These materials or adaptations thereof are not to be sold or otherwise offered for consideration. This Terms of Use slide or page is to be included within the original materials or any adaptations thereof. Boolean Algebra and Logic Gates 94
95 Introduction to Karnaugh Maps If implemented correctly, they almost always produce a minimum solution. They are more straightforward that algebraic manipulations They generally produce SOPs, but POS can be generated from their complements if required.
96 Twovariable Karnaugh maps A B 0 0 B A A B AB A B A B AB A B 0 0 m 0 m 2 m m 3
97 Threevariable Karnaugh maps Product terms corresponding to groups of two.
98 Fourvariable Karnaugh maps
99 Implicants An implicant of a function is a product term. From the point of view of the map, an implicant is a rectangle of, 2, 4, 8,... (any power of 2) s. That rectangle may not include any 0 s. Example: f = A B C D +A B CD+A BCD+AB CD+ABC D +ABC D+ABCD AB CD
100 Implicants The implicants of f are: Minterms ( implicant) A B C D A B CD A BCD ABC D ABC D ABCD AB CD Groups of 2 A CD BCD ACD B CD ABC ABD Groups of 4 CD
101 Prime and Essential Prime Implicants Prime Implicant: o an implicant that (from the point of view of the map) is not fully contained in any one other implicant. Essential Prime Implicant: o a prime implicant that includes at least one that is not included in any other prime implicant. CD AB
102 Minimum SOP Expressions From Karnaugh Maps
103 Minimum SOP Expressions From Karnaugh Maps. Find all essential prime implicants. Circle them on the map and mark the minterm(s) that make them essential with an asterisk (*). 2. Find enough other prime implicants to cover the function. Do this using two criteria: a. Choose a prime implicant that covers as many new s (that is, those not already covered by a chosen prime implicant). b. Avoid leaving isolated uncovered s. The main idea is. To Have all ones covered 2. To Have as few terms as possible 3. To have several rectangles with more s and few rectangles with less s
104 Example f = w x y z +w xy z + w xy z+ w xyz+ wx y z + w xyz+ wxy z + wxyz AB CD * * * 0 unnecessary * 0 f = y z +wyz+w xz
105 f = b + a' c In Groups
106 Prime implicant Don t Cares A rectangle of, 2, 4, 8,... s or X s not included in any one larger rectangle. From the point of view of finding prime implicants, X s (don t cares) are treated as s. Essential prime implicant A prime implicant that covers at least one not covered by any other prime implicant (as always). Don t cares (X s) do not make a prime implicant essential.
107 Example f = Σm(,7,0,,3) + Σd(5,8,5) CD 00 AB x 0 x x 0 Use don t cares to get as many minterms in each tem as possible F = BD + A C D + AB C
108 For the following problem, find the minimum SOP expression within the options given h(a,b,c) = Σm(0,,5) + d(3,4,6,7) In groups a) h = a'b' + c + a b) h = a + c + b c) h = c + b d) h = b e) h = c
109 Implementation of Two Functions
110 F = A B C +A BC +ABC +ABC; G = A B C+A BC+ABC +ABC AB C 0 AB C 0 Example F = A C +AB G = A C+AB
111 Example F = A B + ABC G = A B + BC F = A B + ABC G = A B + ABC
112 Example f = a b + bc g = ab + ac f = a b + abc g = ab + abc
113 Example F = A C + A C D + AB C G = AC + A C D + AB C
114 Try
115 Many electronic systems automatically invert gates Easier to fabricate with electronic components Basic gates used in integrated circuits (IC) digital logic families. NAND gate universal gate Could be used to construct any logic gate NAND, NOR
116 NAND gates. Alternate symbol for NAND. Symbols for NOR gate.
117 NAND Gate Implementation When we have a circuit consisting of AND and OR gates such that. the output of the circuit comes from an OR, 2. the inputs to all OR gates come either from a system input or from the output of an AND, and 3. the inputs to all AND gates come either from a system input or from the output of an OR. All gates are replaced by NAND gates, and any input coming directly into an OR is complemented.
118 Try: g = wx(y+z)+x y Example
119 NOR Gate Implementation When we have a circuit consisting of AND and OR gates such that. the output of the circuit comes from an AND, 2. the inputs to all OR gates come either from a system input or from the output of an AND, and 3. the inputs to all AND gates come either from a system input or from the output of an OR. All gates are replaced by NOR gates, and any input coming directly into an AND is complemented.
120 Example Try: g = (x+y )(x +y)(x +z)
121 XOR and XNOR A xor B is if a = or b is and 0 if both are or 0; Develop a truth table for XOR
122 Homework
Chapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active
More informationChapter 2 Boolean Algebra and Logic Gates
Ch1: Digital Systems and Binary Numbers Ch2: Ch3: GateLevel Minimization Ch4: Combinational Logic Ch5: Synchronous Sequential Logic Ch6: Registers and Counters Switching Theory & Logic Design Prof. Adnan
More informationLogic and Computer Design Fundamentals. Chapter 2 Combinational Logic Circuits. Part 1 Gate Circuits and Boolean Equations
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 28 Pearson Education, Inc. (Hperlinks are active in
More informationECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2
ECE 20B, Winter 2003 Introduction to Electrical Engineering, II LECTURE NOTES #2 Instructor: Andrew B. Kahng (lecture) Email: abk@ucsd.edu Telephone: 8588224884 office, 8583530550 cell Office: 3802
More informationChapter 2 Combinational
Computer Engineering 1 (ECE290) Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations HOANG Trang Reference: 2008 Pearson Education, Inc. Overview Part 1 Gate Circuits and Boolean
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. Overview Part 1 Gate
More informationENG2410 Digital Design Combinational Logic Circuits
ENG240 Digital Design Combinational Logic Circuits Fall 207 S. Areibi School of Engineering University of Guelph Binary variables Binary Logic Can be 0 or (T or F, low or high) Variables named with single
More informationGoals for Lecture. Binary Logic and Gates (MK 2.1) Binary Variables. Notation Examples. Logical Operations
Introduction to Electrical Engineering, II LETURE NOTES #2 Instructor: Email: Telephone: Office: ndrew. Kahng (lecture) abk@ucsd.edu 8588224884 office 3802 P&M lass Website: http://vlsicad.ucsd.edu/courses/ece20b/wi04/
More informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationMC9211 Computer Organization
MC92 Computer Organization Unit : Digital Fundamentals Lesson2 : Boolean Algebra and Simplification (KSB) (MCA) (292/ODD) (29  / A&B) Coverage Lesson2 Introduces the basic postulates of Boolean Algebra
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 1 Gate Circuits and Boolean Equations Chapter 2  Part 1 2 Chapter 2  Part 1 3 Chapter 2  Part 1 4 Chapter 2  Part
More informationCHAPTER III BOOLEAN ALGEBRA
CHAPTER III CHAPTER III CHAPTER III R.M. Dansereau; v.. CHAPTER III2 BOOLEAN VALUES INTRODUCTION BOOLEAN VALUES Boolean algebra is a form of algebra that deals with single digit binary values and variables.
More informationLecture 2 Review on Digital Logic (Part 1)
Lecture 2 Review on Digital Logic (Part 1) Xuan Silvia Zhang Washington University in St. Louis http://classes.engineering.wustl.edu/ese461/ Grading Engagement 5% Review Quiz 10% Homework 10% Labs 40%
More informationChap 2. Combinational Logic Circuits
Overview 2 Chap 2. Combinational Logic Circuits Spring 24 Part Gate Circuits and Boolean Equations Binary Logic and Gates Boolean Algebra Standard Forms Part 2 Circuit Optimization TwoLevel Optimization
More informationCS 121 Digital Logic Design. Chapter 2. Teacher Assistant. Hanin Abdulrahman
CS 121 Digital Logic Design Chapter 2 Teacher Assistant Hanin Abdulrahman 1 2 Outline 2.2 Basic Definitions 2.3 Axiomatic Definition of Boolean Algebra. 2.4 Basic Theorems and Properties 2.5 Boolean Functions
More informationWeekI. Combinational Logic & Circuits
WeekI Combinational Logic & Circuits Overview Binary logic operations and gates Switching algebra Algebraic Minimization Standard forms Karnaugh Map Minimization Other logic operators IC families and
More informationLecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions
EE210: Switching Systems Lecture 5: NAND, NOR and XOR Gates, Simplification of Algebraic Expressions Prof. YingLi Tian Feb. 15, 2018 Department of Electrical Engineering The City College of New York The
More informationLogic Design. Chapter 2: Introduction to Logic Circuits
Logic Design Chapter 2: Introduction to Logic Circuits Introduction Logic circuits perform operation on digital signal Digital signal: signal values are restricted to a few discrete values Binary logic
More informationDigital Logic Design. Combinational Logic
Digital Logic Design Combinational Logic Minterms A product term is a term where literals are ANDed. Example: x y, xz, xyz, A minterm is a product term in which all variables appear exactly once, in normal
More informationLecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps
EE210: Switching Systems Lecture 6: Manipulation of Algebraic Functions, Boolean Algebra, Karnaugh Maps Prof. YingLi Tian Feb. 21/26, 2019 Department of Electrical Engineering The City College of New York
More informationEEE130 Digital Electronics I Lecture #4
EEE130 Digital Electronics I Lecture #4  Boolean Algebra and Logic Simplification  By Dr. Shahrel A. Suandi Topics to be discussed 41 Boolean Operations and Expressions 42 Laws and Rules of Boolean
More informationChapter 2: Switching Algebra and Logic Circuits
Chapter 2: Switching Algebra and Logic Circuits Formal Foundation of Digital Design In 1854 George Boole published An investigation into the Laws of Thoughts Algebraic system with two values 0 and 1 Used
More informationINTRODUCTION TO INFORMATION & COMMUNICATION TECHNOLOGY LECTURE 8 : WEEK 8 CSC110T
INTRODUCTION TO INFORMATION & COMMUNICATION TECHNOLOGY LECTURE 8 : WEEK 8 CSC110T Credit : (2 + 1) / Week TEXT AND REF. BOOKS Text Book: Peter Norton (2011), Introduction to Computers, 7 /e, McGrawHill
More informationII. COMBINATIONAL LOGIC DESIGN.  algebra defined on a set of 2 elements, {0, 1}, with binary operators multiply (AND), add (OR), and invert (NOT):
ENGI 386 Digital Logic II. COMBINATIONAL LOGIC DESIGN Combinational Logic output of digital system is only dependent on current inputs (i.e., no memory) (a) Boolean Algebra  developed by George Boole
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev
E&CE 223 Digital Circuits & Systems Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean Algebra & Logic Gates Major topics Boolean algebra NAND & NOR gates Boolean
More informationChapter 2: Boolean Algebra and Logic Gates
Chapter 2: Boolean Algebra and Logic Gates Mathematical methods that simplify binary logics or circuits rely primarily on Boolean algebra. Boolean algebra: a set of elements, a set of operators, and a
More informationGateLevel Minimization
GateLevel Minimization Dr. Bassem A. Abdullah Computer and Systems Department Lectures Prepared by Dr.Mona Safar, Edited and Lectured by Dr.Bassem A. Abdullah Outline 1. The Map Method 2. Fourvariable
More informationSignals and Systems Digital Logic System
Signals and Systems Digital Logic System Prof. Wonhee Kim Chapter 2 Design Process for Combinational Systems Step 1: Represent each of the inputs and outputs in binary Step 1.5: If necessary, break the
More informationCombinational Logic Fundamentals
Topic 3: Combinational Logic Fundamentals In this note we will study combinational logic, which is the part of digital logic that uses Boolean algebra. All the concepts presented in combinational logic
More informationECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering
ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering Boolean Algebra Boolean Algebra A Boolean algebra is defined with: A set of
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Goal: To obtain the simplest implementation for a given function Optimization is a more formal
More informationELCT201: DIGITAL LOGIC DESIGN
ELCT2: DIGITAL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim Alexan, wassim.joseph@guc.edu.eg Lecture 2 Following the slides of Dr. Ahmed H. Madian ذو الحجة 438 ه Winter
More informationCHAPTER 3 BOOLEAN ALGEBRA
CHAPTER 3 BOOLEAN ALGEBRA (continued) This chapter in the book includes: Objectives Study Guide 3.1 Multiplying Out and Factoring Expressions 3.2 ExclusiveOR and Equivalence Operations 3.3 The Consensus
More informationUNIT 5 KARNAUGH MAPS Spring 2011
UNIT 5 KRNUGH MPS Spring 2 Karnaugh Maps 2 Contents Minimum forms of switching functions Two and threevariable Fourvariable Determination of minimum expressions using essential prime implicants Fivevariable
More informationEC121 Digital Logic Design
EC121 Digital Logic Design Lecture 2 [Updated on 020418] Boolean Algebra and Logic Gates Dr Hashim Ali Spring 2018 Department of Computer Science and Engineering HITEC University Taxila!1 Overview What
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 3 Additional Gates and Circuits Charles Kime & Thomas Kaminski 2008 Pearson Education, Inc. (Hyperlinks are active in
More informationSlide Set 3. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary
Slide Set 3 for ENEL 353 Fall 2016 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2016 SN s ENEL 353 Fall 2016 Slide Set 3 slide
More informationE&CE 223 Digital Circuits & Systems. Lecture Transparencies (Boolean Algebra & Logic Gates) M. Sachdev. Section 2: Boolean Algebra & Logic Gates
Digital Circuits & Systems Lecture Transparencies (Boolean lgebra & Logic Gates) M. Sachdev 4 of 92 Section 2: Boolean lgebra & Logic Gates Major topics Boolean algebra NND & NOR gates Boolean algebra
More informationBinary Logic and Gates. Our objective is to learn how to design digital circuits.
Binary Logic and Gates Introduction Our objective is to learn how to design digital circuits. These circuits use binary systems. Signals in such binary systems may represent only one of 2 possible values
More informationChapter 2: Princess Sumaya Univ. Computer Engineering Dept.
hapter 2: Princess Sumaya Univ. omputer Engineering Dept. Basic Definitions Binary Operators AND z = x y = x y z=1 if x=1 AND y=1 OR z = x + y z=1 if x=1 OR y=1 NOT z = x = x z=1 if x=0 Boolean Algebra
More informationCombinational Logic. Review of Combinational Logic 1
Combinational Logic! Switches > Boolean algebra! Representation of Boolean functions! Logic circuit elements  logic gates! Regular logic structures! Timing behavior of combinational logic! HDLs and combinational
More informationDigital Circuit And Logic Design I. Lecture 3
Digital Circuit And Logic Design I Lecture 3 Outline Combinational Logic Design Principles (). Introduction 2. Switching algebra 3. Combinationalcircuit analysis 4. Combinationalcircuit synthesis Panupong
More informationKarnaugh Maps Objectives
Karnaugh Maps Objectives For Karnaugh Maps of up to 5 variables Plot a function from algebraic, minterm or maxterm form Obtain minimum Sum of Products and Product of Sums Understand the relationship between
More informationUNIVERSITI TENAGA NASIONAL. College of Information Technology
UNIVERSITI TENAGA NASIONAL College of Information Technology BACHELOR OF COMPUTER SCIENCE (HONS.) FINAL EXAMINATION SEMESTER 2 2012/2013 DIGITAL SYSTEMS DESIGN (CSNB163) January 2013 Time allowed: 3 hours
More informationAdministrative Notes. Chapter 2 <9>
Administrative Notes Note: New homework instructions starting with HW03 Homework is due at the beginning of class Homework must be organized, legible (messy is not), and stapled to be graded Chapter 2
More informationLecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University
Lecture 6: Gate Level Minimization Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Aby K George, ECE Department, Wayne State University Contents The Map method Two variable
More informationCombinational Logic Design Principles
Combinational Logic Design Principles Switching algebra Doru Todinca Department of Computers Politehnica University of Timisoara Outline Introduction Switching algebra Axioms of switching algebra Theorems
More informationChapter2 BOOLEAN ALGEBRA
Chapter2 BOOLEAN ALGEBRA Introduction: An algebra that deals with binary number system is called Boolean Algebra. It is very power in designing logic circuits used by the processor of computer system.
More informationEECS150  Digital Design Lecture 19  Combinational Logic Circuits : A Deep Dive
EECS150  Digital Design Lecture 19  Combinational Logic Circuits : A Deep Dive March 30, 2010 John Wawrzynek Spring 2010 EECS150  Lec19cl1 Page 1 Boolean Algebra I (Representations of Combinational
More informationDIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA
DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA 1 Learning Objectives Understand the basic operations and laws of Boolean algebra. Relate these operations and laws to circuits composed of AND gates, OR gates, INVERTERS
More informationLecture 7: Karnaugh Map, Don t Cares
EE210: Switching Systems Lecture 7: Karnaugh Map, Don t Cares Prof. YingLi Tian Feb. 28, 2019 Department of Electrical Engineering The City College of New York The City University of New York (CUNY) 1
More informationBoolean Algebra and Logic Design (Class 2.2 1/24/2013) CSE 2441 Introduction to Digital Logic Spring 2013 Instructor Bill Carroll, Professor of CSE
Boolean Algebra and Logic Design (Class 2.2 1/24/2013) CSE 2441 Introduction to Digital Logic Spring 2013 Instructor Bill Carroll, Professor of CSE Today s Topics Boolean algebra applications in logic
More informationUnit 2 Session  6 Combinational Logic Circuits
Objectives Unit 2 Session  6 Combinational Logic Circuits Draw 3 variable and 4 variable Karnaugh maps and use them to simplify Boolean expressions Understand don t Care Conditions Use the ProductofSums
More informationThis form sometimes used in logic circuit, example:
Objectives: 1. Deriving of logical expression form truth tables. 2. Logical expression simplification methods: a. Algebraic manipulation. b. Karnaugh map (kmap). 1. Deriving of logical expression from
More informationChapter 2. Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates Basic Definitions A binary operator defined on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S. The most common
More informationof Digital Electronics
26 Digital Electronics 729 Digital Electronics 26.1 Analog and Digital Signals 26.3 Binary Number System 26.5 Decimal to Binary Conversion 26.7 Octal Number System 26.9 BinaryCoded Decimal Code (BCD Code)
More informationChapter 2 Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates The most common postulates used to formulate various algebraic structures are: 1. Closure. N={1,2,3,4 }, for any a,b N we obtain a unique c N by the operation
More informationUnit 2 Boolean Algebra
Unit 2 Boolean Algebra 2.1 Introduction We will use variables like x or y to represent inputs and outputs (I/O) of a switching circuit. Since most switching circuits are 2 state devices (having only 2
More information2009 Spring CS211 Digital Systems & Lab CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS
CHAPTER 2: INTRODUCTION TO LOGIC CIRCUITS What will we learn? 2 Logic functions and circuits Boolean Algebra Logic gates and Synthesis CAD tools and VHDL Read Section 2.9 and 2.0 Terminology 3 Digital
More informationChapter 2 Combinational Logic Circuits
Logic and Computer Design Fundamentals Chapter 2 Combinational Logic Circuits Part 2 Circuit Optimization Charles Kime & Thomas Kaminski 2004 Pearson Education, Inc. Terms of Use (Hyperlinks are active
More informationDIGITAL ELECTRONICS & it0203 Semester 3
DIGITAL ELECTRONICS & it0203 Semester 3 P.Rajasekar & C.M.T.Karthigeyan Asst.Professor SRM University, Kattankulathur School of Computing, Department of IT 8/22/2011 1 Disclaimer The contents of the slides
More informationReview for Test 1 : Ch1 5
Review for Test 1 : Ch1 5 October 5, 2006 Typeset by FoilTEX Positional Numbers 527.46 10 = (5 10 2 )+(2 10 1 )+(7 10 0 )+(4 10 1 )+(6 10 2 ) 527.46 8 = (5 8 2 ) + (2 8 1 ) + (7 8 0 ) + (4 8 1 ) + (6 8
More informationCh 2. Combinational Logic. II  Combinational Logic Contemporary Logic Design 1
Ch 2. Combinational Logic II  Combinational Logic Contemporary Logic Design 1 Combinational logic Define The kind of digital system whose output behavior depends only on the current inputs memoryless:
More informationContents. Chapter 2 Digital Circuits Page 1 of 30
Chapter 2 Digital Circuits Page 1 of 30 Contents Contents... 1 2 Digital Circuits... 2 2.1 Binary Numbers... 2 2.2 Binary Switch... 4 2.3 Basic Logic Operators and Logic Expressions... 5 2.4 Truth Tables...
More informationCHAPTER 2 BOOLEAN ALGEBRA
CHAPTER 2 BOOLEAN ALGEBRA This chapter in the book includes: Objectives Study Guide 2.1 Introduction 2.2 Basic Operations 2.3 Boolean Expressions and Truth Tables 2.4 Basic Theorems 2.5 Commutative, Associative,
More informationUNIT 3 BOOLEAN ALGEBRA (CONT D)
UNIT 3 BOOLEAN ALGEBRA (CONT D) Spring 2011 Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions ExclusiveOR and ExclusiveNOR operations The consensus theorem Summary of algebraic
More informationBinary logic consists of binary variables and logical operations. The variables are
1) Define binary logic? Binary logic consists of binary variables and logical operations. The variables are designated by the alphabets such as A, B, C, x, y, z, etc., with each variable having only two
More informationBoolean Algebra and Logic Gates
Boolean Algebra and Logic Gates ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outlines Basic
More informationLogic Gate Level. Part 2
Logic Gate Level Part 2 Constructing Boolean expression from First method: write nonparenthesized OR of ANDs Each AND is a 1 in the result column of the truth table Works best for table with relatively
More informationBoolean Algebra CHAPTER 15
CHAPTER 15 Boolean Algebra 15.1 INTRODUCTION Both sets and propositions satisfy similar laws, which are listed in Tables 11 and 41 (in Chapters 1 and 4, respectively). These laws are used to define an
More informationCombinatorial Logic Design Principles
Combinatorial Logic Design Principles ECGR2181 Chapter 4 Notes Logic System Design I 41 Boolean algebra a.k.a. switching algebra deals with boolean values  0, 1 Positivelogic convention analog voltages
More informationBoolean Algebra and Digital Logic
All modern digital computers are dependent on circuits that implement Boolean functions. We shall discuss two classes of such circuits: Combinational and Sequential. The difference between the two types
More informationWhy digital? Overview. Number Systems. Binary to Decimal conversion
Why digital? Overview It has the following advantages over analog. It can be processed and transmitted efficiently and reliably. It can be stored and retrieved with greater accuracy. Noise level does not
More informationBoolean Algebra & Logic Gates. By : Ali Mustafa
Boolean Algebra & Logic Gates By : Ali Mustafa Digital Logic Gates There are three fundamental logical operations, from which all other functions, no matter how complex, can be derived. These Basic functions
More informationCHAPTER1: Digital Logic Circuits Combination Circuits
CS224: Computer Organization S.KHABET CHAPTER1: Digital Logic Circuits Combination Circuits 1 PRIMITIVE LOGIC GATES Each of our basic operations can be implemented in hardware using a primitive logic gate.
More informationSimplifying Logic Circuits with Karnaugh Maps
Simplifying Logic Circuits with Karnaugh Maps The circuit at the top right is the logic equivalent of the Boolean expression: f = abc + abc + abc Now, as we have seen, this expression can be simplified
More informationELC224C. Karnaugh Maps
KARNAUGH MAPS Function Simplification Algebraic Simplification Half Adder Introduction to Kmaps How to use Kmaps Converting to Minterms Form Prime Implicants and Essential Prime Implicants Example on
More informationUNIT 4 MINTERM AND MAXTERM EXPANSIONS
UNIT 4 MINTERM AND MAXTERM EXPANSIONS Spring 2 Minterm and Maxterm Expansions 2 Contents Conversion of English sentences to Boolean equations Combinational logic design using a truth table Minterm and
More informationCS 226: Digital Logic Design
CS 226: Digital Logic Design 0 1 1 I S 0 1 0 S Department of Computer Science and Engineering, Indian Institute of Technology Bombay. 1 of 29 Objectives In this lecture we will introduce: 1. Logic functions
More information211: Computer Architecture Summer 2016
211: Computer Architecture Summer 2016 Liu Liu Topic: Storage Project3 Digital Logic  Storage: Recap  Review: cache hit rate  Project3  Digital Logic:  truth table => SOP  simplification: Boolean
More informationStandard Expression Forms
ThisLecture will cover the following points: Canonical and Standard Forms MinTerms and MaxTerms Digital Logic Families 24 March 2010 Standard Expression Forms Two standard (canonical) expression forms
More informationFunctions. Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways:
Boolean Algebra (1) Functions Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: An expression is finite but not unique f(x,y)
More informationChapter 2 : Boolean Algebra and Logic Gates
Chapter 2 : Boolean Algebra and Logic Gates By Electrical Engineering Department College of Engineering King Saud University 14311432 2.1. Basic Definitions 2.2. Basic Theorems and Properties of Boolean
More informationMidterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes
Page 1 of 10 School of Computer Science 6026501 Computer Architecture and Digital Design Winter 2009 Semester Midterm Examination # 1 Wednesday, February 25, 2009 Student Name: First Name Family Name
More informationDigital Logic Design. Malik Najmus Siraj
Digital Logic Design Malik Najmus Siraj siraj@case.edu.pkedu LECTURE 4 Today s Agenda Recap 2 s complement Binary Logic Boolean algebra Recap Computer Arithmetic Signed numbers Radix and diminished radix
More informationUNIT 1. BOOLEAN ALGEBRA AND COMBINATIONAL CIRCUITS
UNIT 1. BOOLEAN ALGEBRA AND COMBINATIONAL CIRCUITS Numerical Presentation: In science, technology, business, and, in fact, most other fields of endeavour, we are constantly dealing with quantities. Quantities
More informationL2: Combinational Logic Design (Construction and Boolean Algebra)
L2: Combinational Logic Design (Construction and Boolean Algebra) Acknowledgements: Lecture material adapted from Chapter 2 of R. Katz, G. Borriello, Contemporary Logic Design (second edition), Pearson
More informationCHAPTER 7. Exercises 17/ / /2 2 0
CHAPTER 7 Exercises E7. (a) For the whole part, we have: Quotient Remainders 23/2 /2 5 5/2 2 2/2 0 /2 0 Reading the remainders in reverse order, we obtain: 23 0 = 0 2 For the fractional part we have 2
More informationPart 1: Digital Logic and Gates. Analog vs. Digital waveforms. The digital advantage. In real life...
Part 1: Digital Logic and Gates Analog vs Digital waveforms An analog signal assumes a continuous range of values: v(t) ANALOG A digital signal assumes discrete (isolated, separate) values Usually there
More informationChapter 2 Combinational logic
Chapter 2 Combinational logic Chapter 2 is very easy. I presume you already took discrete mathemtics. The major part of chapter 2 is boolean algebra. II  Combinational Logic Copyright 24, Gaetano Borriello
More informationChapter 2 Boolean Algebra and Logic Gates
Chapter 2 Boolean Algebra and Logic Gates Huntington Postulates 1. (a) Closure w.r.t. +. (b) Closure w.r.t.. 2. (a) Identity element 0 w.r.t. +. x + 0 = 0 + x = x. (b) Identity element 1 w.r.t.. x 1 =
More informationUnit 2 Boolean Algebra
Unit 2 Boolean Algebra 1. Developed by George Boole in 1847 2. Applied to the Design of Switching Circuit by Claude Shannon in 1939 Department of Communication Engineering, NCTU 1 2.1 Basic Operations
More informationChapter 2 Boolean Algebra and Logic Gates
CSA051  Digital Systems 數位系統導論 Chapter 2 Boolean Algebra and Logic Gates 吳俊興國立高雄大學資訊工程學系 Chapter 2. Boolean Algebra and Logic Gates 21 Basic Definitions 22 Axiomatic Definition of Boolean Algebra 23
More informationCOS 140: Foundations of Computer Science
COS 140: Foundations of Computer Science Boolean Algebra Fall 2018 Introduction 3 Problem................................................................. 3 Boolean algebra...........................................................
More informationEvery time has a value associated with it, not just some times. A variable can take on any value within a range
Digital Logic Circuits Binary Logic and Gates Logic Simulation Boolean Algebra NAND/NOR and XOR gates Decoder fundamentals Half Adder, Full Adder, Ripple Carry Adder Analog vs Digital Analog Continuous»
More informationBOOLEAN ALGEBRA. Introduction. 1854: Logical algebra was published by George Boole known today as Boolean Algebra
BOOLEAN ALGEBRA Introduction 1854: Logical algebra was published by George Boole known today as Boolean Algebra It s a convenient way and systematic way of expressing and analyzing the operation of logic
More informationBinary Logic and Gates
1 COE 202 Digital Logic Binary Logic and Gates Dr. Abdulaziz Y. Barnawi COE Department KFUPM 2 Outline Introduction Boolean Algebra Elements of Boolean Algebra (Binary Logic) Logic Operations & Logic
More informationAdvanced Digital Design with the Verilog HDL, Second Edition Michael D. Ciletti Prentice Hall, Pearson Education, 2011
Problem 21 Recall that a minterm is a cube in which every variable appears. A Boolean expression in SOP form is canonical if every cube in the expression has a unique representation in which all of the
More informationChapter 3. Boolean Algebra. (continued)
Chapter 3. Boolean Algebra (continued) Algebraic structure consisting of: set of elements B binary operations {+, } unary operation {'} Boolean Algebra such that the following axioms hold:. B contains
More informationSimplification of Boolean Functions. Dept. of CSE, IEM, Kolkata
Simplification of Boolean Functions Dept. of CSE, IEM, Kolkata 1 Simplification of Boolean Functions: An implementation of a Boolean Function requires the use of logic gates. A smaller number of gates,
More information