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The document discusses different implementations of Boolean functions including NAND, AND-OR-INVERT, and NOR. It provides examples of converting binary numbers to decimal, hexadecimal, and octal representations.
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R. Kime Section 2 – Fall 2001 Chapter 2 – Combinational Logic Circuits – Part 7 Charles Kime
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and Computer Design Fundamentals © 2001 Prentice Hall, Inc Chapter 2-Part 7 2 NAND and
NOR Implementation We found that we could implement general Boolean equations with these
three primitives: • AND • OR • NOT In this section we will find that either of two gates, the
NAND gate or the NOR gate can be used to implement arbitrary logic functions. We use the
Positive Logic Convention (where all signals are active high) and a small circle to on a symbol to
represent NOT or invert. 2 Logic and Computer Design Fundamentals © 2001 Prentice Hall, Inc
Chapter 2-Part 7 3 NAND Gates The basic positive logic NAND gate is denoted by the
following symbol: • AND-Invert (NAND) NAND comes from NOT AND, I. e., the AND
function with a NOT applied. We call this symbol for a NAND gate an AND-Invert. The small
circle represents the invert function. If we apply DeMorgan\'s Law we get: X Y Z X YZ = X +
Y + Z F(X, Y,Z) = X Y Z Logic and Computer Design Fundamentals © 2001 Prentice Hall, Inc
Chapter 2-Part 7 4 NAND Gates (Cont.) Applying DeMorgan\'s Law gives: • Invert-OR
(NAND) We call this symbol for a NAND gate the Invert - OR since all inputs are inverted,
followed by the OR function. Both symbols represent the NAND gate - it is sometimes more
logically descriptive to use one form over the other. A NAND gate with one input degenerates to
an inverter. X Y Z F(X, Y,Z) = X + Y + Z 3 Logic and Computer Design Fundamentals © 2001
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Products form. 4 Logic and Computer Design Fundamentals © 2001 Prentice Hall, Inc Chapter
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The document discusses different logic gate implementations using NAND and NOR gates. It explains that AND, OR and NOT functions can be represented as equivalent NAND and NOR logic diagrams. It provides examples of how NAND and NOR gates can be used to implement SUM-OF-PRODUCTS logic functions. Specifically, it shows how a NAND gate implementation can be easily converted to a sum-of-products form using De Morgan's theorem. It also gives examples of NOR gate implementations, as well as how AND-OR-INVERT functions can be implemented using both NAND-AND and AND-OR equivalent forms.
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1 University of Wisconsin - Madison ECE/Comp Sci 352 Digital Systems Fundamentals Charles
R. Kime Section 2 – Fall 2001 Chapter 2 – Combinational Logic Circuits – Part 7 Charles Kime
& Thomas Kaminski © 2001 Prentice Hall, Inc Logic and Computer Design Fundamentals Logic
and Computer Design Fundamentals © 2001 Prentice Hall, Inc Chapter 2-Part 7 2 NAND and
NOR Implementation We found that we could implement general Boolean equations with these
three primitives: • AND • OR • NOT In this section we will find that either of two gates, the
NAND gate or the NOR gate can be used to implement arbitrary logic functions. We use the
Positive Logic Convention (where all signals are active high) and a small circle to on a symbol to
represent NOT or invert. 2 Logic and Computer Design Fundamentals © 2001 Prentice Hall, Inc
Chapter 2-Part 7 3 NAND Gates The basic positive logic NAND gate is denoted by the
following symbol: • AND-Invert (NAND) NAND comes from NOT AND, I. e., the AND
function with a NOT applied. We call this symbol for a NAND gate an AND-Invert. The small
circle represents the invert function. If we apply DeMorgan\'s Law we get: X Y Z X YZ = X +
Y + Z F(X, Y,Z) = X Y Z Logic and Computer Design Fundamentals © 2001 Prentice Hall, Inc
Chapter 2-Part 7 4 NAND Gates (Cont.) Applying DeMorgan\'s Law gives: • Invert-OR
(NAND) We call this symbol for a NAND gate the Invert - OR since all inputs are inverted,
followed by the OR function. Both symbols represent the NAND gate - it is sometimes more
logically descriptive to use one form over the other. A NAND gate with one input degenerates to
an inverter. X Y Z F(X, Y,Z) = X + Y + Z 3 Logic and Computer Design Fundamentals © 2001
Prentice Hall, Inc Chapter 2-Part 7 5 NAND Function Implementation NAND gates can
implement a simplified Sum-ofProducts form. Constructing two level NAND-NAND gate
circuit: The first level is two 2-input NAND gates using ANDInvert. The second level is one 2-
input NAND gate using Invert-OR. Using the NAND relationship, we have: G(A,B,C,D) = AB
CD = AB + CD = A B + CD A B C D G(A,B,C,D) = AB + CD Logic and Computer Design
Fundamentals © 2001 Prentice Hall, Inc Chapter 2-Part 7 6 NAND Implementation (Cont.) In
the implementation, note that the bubbles are on opposite ends of the same line. Thus, they can
be combined and deleted: A B C D G(A,B,C,D) This form of the implementation is the Sum-of-
Products form. 4 Logic and Computer Design Fundamentals © 2001 Prentice Hall, Inc Chapter
2-Part 7 7 NAND Implementation (Cont.) In the implementation, the bubbles are on opposite
ends of the same line. By , they can be combined and deleted: A sum-of-products (SOP) form
results To implement an equation like: F(A,B,C) = A + BC, the NAND for A degenerates t.
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This document provides an overview of Boolean algebra and logic expressions. It covers topics such as:
- Boolean operations like AND, OR, NOT
- Boolean variables, literals, and expressions
- Laws of Boolean algebra including commutative, associative, distributive, and DeMorgan's theorems
- Standard forms of Boolean expressions including sum of products (SOP) and product of sums (POS)
- Converting between Boolean expressions and truth tables
The document is intended to teach the basic concepts and tools used for analyzing and simplifying digital logic circuits and Boolean functions.
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This document discusses various techniques for simplifying Boolean functions including K-maps, don't care conditions, and implementing Boolean functions as logic circuits. It covers:
1) Using K-maps to solve 3-variable functions and simplify sums.
2) How don't care conditions can be represented on K-maps to simplify structures.
3) Converting Boolean functions to logic diagrams using only NAND or NOR gates as universal gates. Steps are provided to convert functions to NAND and NOR gate implementations.
4) Equivalents for NOT, AND and OR gates using only NAND or NOR gates.
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This document provides an overview of Boolean algebra, including its basic operations, laws, and applications to digital logic circuits. Some key points:
- Boolean algebra uses binary operations like AND, OR, and NOT to represent logical relationships between variables that can only have true or false values.
- It has commutative, distributive, complement, and identity laws that allow simplifying logical expressions.
- Boolean algebra is used to analyze logic circuits built from gates like AND, OR, NOT, NAND, and NOR. Truth tables define the output of a circuit for all input combinations.
- Expressions can be converted between sum-of-products and product-of-sums standard forms for analysis and simpl
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This document discusses logic simplification using Karnaugh maps. It begins with an overview of Boolean algebra simplification techniques. It then covers standard forms such as sum-of-products (SOP) and product-of-sums (POS), and how to convert between different forms. The document also discusses mapping logic expressions to Karnaugh maps and using K-map rules for simplification. Truth tables and determining logic expressions from truth tables are also covered.
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The document provides information on logic minimization techniques including Karnaugh maps (K-maps) for two, three, four and more variables, prime implicant charts, don't care conditions, NAND and NOR gate implementations, and the Quine-McCluskey (Q-M) tabulation method. Examples are given for each topic to demonstrate how to use the techniques to minimize logic functions and implement them using basic gates.
14 Lec11 2003
1) The document discusses different types of logic gates and their truth tables. It describes logic functions like AND, OR, NOT, NAND, NOR, and XOR.
2) It explains how to implement logic functions using logic gates by writing the function as a sum of products and then building a circuit from that.
3) DeMorgan's laws allow any logic function to be implemented using only NAND gates. The document discusses how to simplify logic circuits to make them faster and more efficient.
Boolean algebra
Logic gates form the basic building blocks of digital circuits and logic. The three fundamental logic gates are AND, OR, and NOT. NAND and NOR gates are also commonly used as they are universal gates that can be combined to perform all possible logic functions. Techniques for minimizing logic expressions include Karnaugh maps, which allow visualization of minterms, and the Quine-McCluskey method, which systematically finds prime implicants. Implementation of logic functions typically uses NAND or NOR gates due to their simplicity and universality.
Logic Simplification Using SOP
This document discusses simplifying Boolean expressions using Boolean algebra. It explains how to simplify expressions by applying rules like distribution, idempotency, etc. It also covers converting expressions to standard forms, including sum-of-products (SOP) and product-of-sums (POS). Standard forms make expressions easier to evaluate, simplify and implement. The document provides examples of simplifying expressions and converting between SOP and POS form.
abdullah
DeMorgan's theorems state that the complement of a product of variables is equal to the sum of the complemented variables and the complement of a sum of variables is equal to the product of the complemented variables. Boolean expressions can be written in sum-of-products (SOP) form or product-of-sums (POS) form, where in SOP two or more product terms are summed and in POS two or more product terms are multiplied.
e CAD lab manual
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Venkatraman G is seeking an opportunity to improve his skills and knowledge through work at an organization that supports growth. He holds an MCA from Thanthai Hans Roever College with 65.7% and a B.Sc. in Computer Science from the same institution with 63.5%. His skills include languages like MS Office, operating systems like Windows, and web development technologies like HTML, CSS, JavaScript, PHP and MySQL. His past project involved developing a job portal using PHP and MySQL to allow employers to post job vacancies and conduct walking interviews.
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1) Binary addition and subtraction follow similar rules to decimal with 0+0=0, 0+1=1, 1+0=1, 1+1=10, and 1+1+1=11. To subtract larger binary numbers, subtract column by column.
2) 1's complement changes each 0 to 1 and vice versa, and is used to subtract by adding the 1's complement of the subtrahend to the minuend. 2's complement is 1's complement plus 1.
3) 2's complement subtraction discards any end around carry, while 1's and 2's complement allow binary subtraction. 9's and 10's complement perform similar operations in decimal.
Digital
The document discusses several topics in digital logic including:
1) The don't care condition rule for Karnaugh maps which involves treating don't care terms as 1s to find the largest groups.
2) Sum of products and product of sums expressions for logic functions.
3) Truth tables and diagrams for logic gates.
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5) Multiplexers and demultiplexers - how control signals select inputs and outputs.
Digital
The document discusses different implementations of Boolean functions including NAND, AND-OR-INVERT, and NOR. It provides examples of converting binary numbers to decimal, hexadecimal, and octal representations.
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The document discusses different implementations of Boolean functions including NAND, AND-OR-INVERT, and NOR. It provides examples of converting binary numbers to decimal, hexadecimal, and octal representations.
digital Electronics
The document discusses the don't care condition rule for Karnaugh maps. It states that don't cares should be treated as 1s and included in the largest groups when minimizing logic expressions. An example problem with a Karnaugh map is given showing don't cares treated as 1s and the minimized sum of products expression. Truth tables, logic diagrams, multiplexers, demultiplexers and binary coded decimal adders are also briefly discussed.
digital
The document discusses the don't care condition rule for Karnaugh maps. It states that don't cares should be treated as 1s and included in the largest groups when minimizing logic expressions. An example problem with a Karnaugh map is given showing don't cares treated as 1s and the minimized sum of products expression. Truth tables, logic diagrams, multiplexers, demultiplexers and binary coded decimal adders are also briefly discussed.
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1) Binary addition and subtraction follow similar rules to decimal with 0+0=0, 0+1=1, 1+0=1, 1+1=10, and 1+1+1=11. To subtract larger binary numbers, subtract column by column.
2) 1's complement changes each 0 to 1 and vice versa, and is used to subtract by adding the 1's complement of the subtrahend to the minuend. 2's complement is 1's complement plus 1.
3) 2's complement subtraction discards any end around carry, while 1's and 2's complement allow binary subtraction. 9's and 10's complement work similarly to convert decimals.
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Positive logic defines the high voltage level as 1 and the low voltage level as 0. Negative logic defines the high voltage level as 0 and the low voltage level as 1. Logic gates like NAND and NOR will have different truth tables under positive and negative logic systems. The characteristics of a logic family include speed, fan-in, fan-out, noise immunity, and power dissipation. Speed considers transition times, fan-in is the number of inputs, fan-out is the maximum number of output circuits, noise immunity is the tolerance for noise, and power dissipation should be minimized.
Rathika i bca
Boolean algebra is an algebraic structure defined by two binary operators, addition '+' and multiplication '.'. It has identity elements 0 for addition and 1 for multiplication. The variables in Boolean equations can only be 1 or 0. Logical addition follows the OR truth table, logical multiplication the AND truth table, and negation the NOT truth table. These define the Boolean operations of OR, AND and NOT respectively.
Rathika i bca
Boolean algebra is an algebraic structure with two binary operators, addition '+' and multiplication '.'. It is defined on a set of elements where: (1) the results of '+' and '.' are closed within the set of elements, (2) there is an identity element 0 for '+' and an identity element 1 for '.', and (3) the variables can only be 1 or 0. The rules of logical addition, multiplication and negation define OR, AND and NOT operations respectively. For example, logical addition works like an OR operation where 0+0=0 and 1+1=1, while logical multiplication works like an AND operation where 0.0=0 and 1.1=1.
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Sukku
The document discusses different implementations of Boolean functions including NAND, AND-OR-INVERT, and NOR. It provides examples of converting binary numbers to decimal, hexadecimal, and octal representations.
Don’t care contion
The document discusses the don't care condition rule for Karnaugh maps. It states that don't cares should be treated as 1s and included in the largest groups when minimizing logic expressions. An example problem with a Karnaugh map containing don't care terms is shown. Basic concepts involving sum of products, product of sums, truth tables, and logic diagrams for multiplexers, demultiplexers, and BCD adders are also defined.
Presentation1
This document discusses three applications of XOR gates: 1) Using XOR gates to check the parity of words by outputting 0 for even parity and 1 for odd parity. 2) Converting a 6-bit binary input word to a 6-bit Gray code output word. 3) Using a controlled inverter gate that either inverts or does not invert its input depending on the value of an additional control input.
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- 1. NAND Implementation The implementation of a Boolean function with NAND gates requires that the function be simplified in the sum-of-products form. Consider the Boolean function F = AB + CD + E