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
- Boolean algebra uses binary values (1/0) to represent true/false in digital circuits.
- The basic Boolean operations are AND, OR, and NOT. Truth tables and Boolean expressions can both be used to represent the functions of circuits.
- Boolean expressions can be simplified using algebraic rules like commutative, distributive, DeMorgan's, and absorption laws. This allows simpler circuit implementations.
The document discusses Boolean expressions and their use in computer programming. It defines Boolean expressions as expressions that evaluate to true or false. Boolean expressions are composed of logical operators like AND, OR, and NOT. The document then discusses different logical operators and their truth tables. It also covers Boolean algebra identities and theorems. Finally, it introduces concepts like minterms, maxterms, sum of products, and product of sums and how Karnaugh maps can be used to simplify Boolean expressions.
The document discusses different methods for representing signed binary numbers:
1) Sign-magnitude notation represents positive and negative numbers by using the most significant bit to indicate the sign (0 for positive, 1 for negative) and the remaining bits for the magnitude.
2) One's complement represents negative numbers by inverting all bits of the positive number.
3) Two's complement, the most common method, represents negative numbers by inverting all bits and adding 1 to the result. This allows simple addition to perform subtraction.
- Karnaugh maps are used to simplify Boolean algebra expressions by grouping adjacent 1s in a two-dimensional grid.
- Groups must contain powers of 2 cells and cannot include any 0s. They can overlap and wrap around the map.
- The simplified expression is obtained by determining which variables stay the same within each group.
The document discusses simplifying Boolean expressions using Boolean algebra. It provides examples of simplifying the expression AB+A(B+C)+B(B+C) through applying distribution and other Boolean algebra rules until reaching the simplified expression B+AC. The document also discusses standard forms such as sum-of-products (SOP) and product-of-sums (POS), and how to convert between them using techniques such as ensuring all variables appear in each term. Truth tables are presented as a way to represent Boolean expressions and determine if an expression is in standard form.
This document discusses canonical forms for representing Boolean functions. It defines sum of products (SOP) and product of sums (POS) forms, which are standard representations. Minterms and maxterms are also defined as product and sum terms involving all variables. The canonical SOP form is defined as the logical sum of minterms where the function value is 1. Canonical POS form is the logical product of maxterms where the function value is 0. Procedures to convert between SOP, POS and canonical forms are presented. Canonical forms provide a unique representation and can be used to determine equivalency between functions.
- Boolean algebra uses binary values (1/0) to represent true/false in digital circuits.
- The basic Boolean operations are AND, OR, and NOT. Truth tables and Boolean expressions can both be used to represent the functions of circuits.
- Boolean expressions can be simplified using algebraic rules like commutative, distributive, DeMorgan's, and absorption laws. This allows simpler circuit implementations.
The document discusses Boolean expressions and their use in computer programming. It defines Boolean expressions as expressions that evaluate to true or false. Boolean expressions are composed of logical operators like AND, OR, and NOT. The document then discusses different logical operators and their truth tables. It also covers Boolean algebra identities and theorems. Finally, it introduces concepts like minterms, maxterms, sum of products, and product of sums and how Karnaugh maps can be used to simplify Boolean expressions.
The document discusses different methods for representing signed binary numbers:
1) Sign-magnitude notation represents positive and negative numbers by using the most significant bit to indicate the sign (0 for positive, 1 for negative) and the remaining bits for the magnitude.
2) One's complement represents negative numbers by inverting all bits of the positive number.
3) Two's complement, the most common method, represents negative numbers by inverting all bits and adding 1 to the result. This allows simple addition to perform subtraction.
- Karnaugh maps are used to simplify Boolean algebra expressions by grouping adjacent 1s in a two-dimensional grid.
- Groups must contain powers of 2 cells and cannot include any 0s. They can overlap and wrap around the map.
- The simplified expression is obtained by determining which variables stay the same within each group.
The document discusses simplifying Boolean expressions using Boolean algebra. It provides examples of simplifying the expression AB+A(B+C)+B(B+C) through applying distribution and other Boolean algebra rules until reaching the simplified expression B+AC. The document also discusses standard forms such as sum-of-products (SOP) and product-of-sums (POS), and how to convert between them using techniques such as ensuring all variables appear in each term. Truth tables are presented as a way to represent Boolean expressions and determine if an expression is in standard form.
This document discusses canonical forms for representing Boolean functions. It defines sum of products (SOP) and product of sums (POS) forms, which are standard representations. Minterms and maxterms are also defined as product and sum terms involving all variables. The canonical SOP form is defined as the logical sum of minterms where the function value is 1. Canonical POS form is the logical product of maxterms where the function value is 0. Procedures to convert between SOP, POS and canonical forms are presented. Canonical forms provide a unique representation and can be used to determine equivalency between functions.
This document discusses digital arithmetic circuits. It defines competencies around binary addition, 1's and 2's complement representations, and arithmetic operations. Key concepts covered include half adders, full adders, parallel addition, and 8-bit addition. Worked examples demonstrate converting between decimal and binary representations, performing addition and subtraction using 2's complement, and designing multi-bit adders.
Boolean algebra and logic circuits were introduced. Boolean algebra uses binary numbers (0,1) and logical operations like AND, OR, and NOT to simplify logic expressions. Basic logic gates like AND, OR, and NOT were explained. Logic circuits can be built using combinations of logic gates to perform complex logical functions. Boolean algebra is used to simplify logic circuits and increase the efficiency of digital devices like computers.
The document discusses minimizing Boolean expressions using Karnaugh maps. It explains that Karnaugh maps provide a graphical way to simplify logic circuits by grouping adjacent 1s in the map. The steps for minimization using Karnaugh maps are outlined, including drawing the map, entering values, forming the largest possible groups of 1s, and selecting the fewest groups needed to cover all 1s. Rules for grouping such as group size and overlap are also covered.
The document discusses Karnaugh maps (K-maps), which are a tool for representing and simplifying Boolean functions with up to six variables. K-maps arrange the variables in a grid according to their binary values. Adjacent cells that differ in only one variable can be combined to simplify the function by eliminating that variable. The document provides examples of using K-maps to minimize Boolean functions in sum of products and product of sums form. It also discusses techniques like combining cells into the largest groups possible and handling don't-care conditions to further simplify expressions.
Boolean algebra is an algebra of logic developed by George Boole between 1815-1864 to represent logical statements as an algebra of true and false. It is used to perform logical operations in digital computers by representing true as 1 and false as 0. The fundamental logical operators are AND, OR, and NOT. Boolean algebra expressions can be represented in sum of products (SOP) form or product of sums (POS) form and minimized using algebraic rules or Karnaugh maps. Minterms and maxterms are used to derive Boolean functions from truth tables in canonical SOP or POS form.
Fixed-point and floating-point numbers can be represented in computers using binary numbers. Floating-point numbers represent numbers in scientific notation with a sign, mantissa, and exponent. In 8-bit floating point, numbers use 1 bit for sign, 3 bits for exponent, and 4 bits for mantissa, such as 0.001 x 21 = 2.25. Larger precision formats such as 32-bit and 64-bit floating point according to the IEEE standard use more bits for exponent and mantissa.
A combinational circuit is a logic circuit whose output is solely determined by the present input. It has no internal memory and its output depends only on the current inputs. A half adder is a basic combinational circuit that adds two single bits and produces a sum and carry output. A full adder adds three bits and produces a sum and carry like the half adder. Other combinational circuits discussed include half and full subtractors, decoders, encoders, and priority encoders.
Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
This document discusses binary subtraction and multiplication. It provides 4 rules for binary subtraction: 0-0=0, 0-1=1, 1-0=1, and 1-1=0. An example of binary subtraction is shown subtracting 10010 from 10110 with a difference of 00100. Binary multiplication is also covered, noting the multiplier is always 1 or 0, and the 4 basic rules are: 0x0=0, 0x1=0, 1x0=0, and 1x1=1. An example of binary multiplication is shown multiplying 10001111 by 1101000 to get a product of 1101.
Digital systems use logic gates like AND, OR, NOT, NAND, NOR, EXOR and EXNOR. The AND gate outputs 1 only if all inputs are 1. The OR gate outputs 1 if one or more inputs are 1. The NOT gate inverts the input. NAND and NOR gates are combinations of AND/OR gates with NOT gates. EXOR outputs 1 if either but not both inputs are 1, while EXNOR does the opposite. Truth tables define the output of each gate combination.
In electronics, an adder is a digital circuit that performs addition of numbers.
In modern computers and other kinds of processors, adders are used in the arithmetic logic unit (ALU), but also in other parts of the processor, where they are used to calculate addresses, table indices, and similar operations.
This document defines and classifies different types of binary codes. It explains that binary codes represent numeric and alphanumeric data as groups of bits. Binary codes are classified as weighted or non-weighted, reflective or non-reflective, and sequential or non-sequential. Common binary codes include ASCII, EBCDIC, Hollerith, BCD, excess-3, and Gray codes. Error detecting and correcting codes are also discussed which add extra bits to detect or correct errors during data transmission. Examples of different binary codes are provided.
This document provides an outline for a course on digital logic design. It includes the course title and credit hours, topics that will be covered such as Boolean algebra, logic gates, combinational and sequential circuits, programmable logic devices, and memory. It also lists recommended textbooks and provides the grading breakdown. Examples of analogue and digital quantities, signals, and number systems are given. Common logic gates such as AND, OR, NOT, NAND and NOR are described along with their truth tables and applications. Combinational circuits, functional devices, sequential circuits and memory are also introduced.
The document discusses binary subtraction and different types of binary subtractors. It describes half subtractors and full subtractors. A half subtractor is a basic circuit that can subtract two binary bits and outputs the difference and borrow. A full subtractor can subtract three bits by also considering the borrow from the previous stage. Truth tables and K-maps are used to derive the logic equations for difference and borrow outputs. Full subtractors are realized using basic gates by complementing one input to convert a full adder circuit into a full subtractor.
presentation on (Boolean rules & laws)kinza arshad
This document contains a presentation outline on Boolean algebra that was presented to a professor. The presentation covered the rules of Boolean algebra, laws of Boolean algebra, and examples of simplifying Boolean expressions. Some key points include:
- The rules of Boolean algebra include basic logical operations like A+1=1 and A.A=0.
- The laws include commutative, associative, and distributive laws that govern how logical operations are rearranged.
- Examples showed simplifying expressions using the rules and laws, like applying distribution to break apart expressions.
- Truth tables were included to verify the logical equivalencies hold for all possible variable values.
Binary arithmetic is essential for digital computers and systems. It involves adding, subtracting, multiplying, and dividing binary numbers using basic rules. Signed binary numbers represent positive and negative values using sign-magnitude, 1's complement, and 2's complement methods. Arithmetic operations on signed binary numbers follow rules for handling the sign bit and complement representations.
1) Karnaugh maps provide a systematic method for simplifying Boolean expressions and minimizing them to their simplest forms.
2) Karnaugh maps arrange variables in a two-dimensional grid where each cell represents a minterm and adjacent cells differ in only one variable.
3) Expressions can be minimized by grouping adjacent cells containing 1s and eliminating any variables that change across the group's boundaries.
Karnaugh maps (K-maps) are used to simplify Boolean logic expressions. A K-map arranges the minterms from a truth table into an array of cells where adjacent cells differ by only one variable. Groups of adjacent 1s in the K-map correspond to terms in a sum-of-products expression. The process of mapping a logic function onto a K-map and grouping 1s results in a minimum simplified expression. Don't care conditions can be treated as 1s to form larger groups for greater simplification. Both sum-of-products and product-of-sums expressions can be mapped and minimized using K-maps.
1. Boolean algebra is a mathematical system used to specify and transform logic functions. It uses binary variables that take on values of 1 or 0 and logical operators like AND, OR, and NOT.
2. Logic gates implement logic functions physically using electronic components. Common gates are AND, OR, and NOT. Gates have a small but nonzero delay between input change and output change.
3. Boolean expressions can be represented using truth tables, logic diagrams, or algebraic expressions. Standard forms include sum-of-minterms and product-of-maxterms forms.
The document discusses binary arithmetic operations including addition, subtraction, multiplication, and division. It provides examples and step-by-step explanations of how to perform each operation in binary. For addition and subtraction, it explains the rules and concepts like carry bits and two's complement. For multiplication, it describes the shift-and-add method. And for division, it outlines the long division approach of shift-and-subtract in binary.
Lecture 05-Logic expression and Boolean Algebra.pptxWilliamJosephat1
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.
This document provides an overview of Boolean algebra and logic gates. It introduces Boolean logic operations like AND, OR, and NOT. It covers Boolean algebra laws and De Morgan's theorems. It also discusses logic gate types like AND, OR, NOT, NAND, NOR, XOR and XNOR. Karnaugh maps are introduced as a method to simplify Boolean expressions.
This document discusses digital arithmetic circuits. It defines competencies around binary addition, 1's and 2's complement representations, and arithmetic operations. Key concepts covered include half adders, full adders, parallel addition, and 8-bit addition. Worked examples demonstrate converting between decimal and binary representations, performing addition and subtraction using 2's complement, and designing multi-bit adders.
Boolean algebra and logic circuits were introduced. Boolean algebra uses binary numbers (0,1) and logical operations like AND, OR, and NOT to simplify logic expressions. Basic logic gates like AND, OR, and NOT were explained. Logic circuits can be built using combinations of logic gates to perform complex logical functions. Boolean algebra is used to simplify logic circuits and increase the efficiency of digital devices like computers.
The document discusses minimizing Boolean expressions using Karnaugh maps. It explains that Karnaugh maps provide a graphical way to simplify logic circuits by grouping adjacent 1s in the map. The steps for minimization using Karnaugh maps are outlined, including drawing the map, entering values, forming the largest possible groups of 1s, and selecting the fewest groups needed to cover all 1s. Rules for grouping such as group size and overlap are also covered.
The document discusses Karnaugh maps (K-maps), which are a tool for representing and simplifying Boolean functions with up to six variables. K-maps arrange the variables in a grid according to their binary values. Adjacent cells that differ in only one variable can be combined to simplify the function by eliminating that variable. The document provides examples of using K-maps to minimize Boolean functions in sum of products and product of sums form. It also discusses techniques like combining cells into the largest groups possible and handling don't-care conditions to further simplify expressions.
Boolean algebra is an algebra of logic developed by George Boole between 1815-1864 to represent logical statements as an algebra of true and false. It is used to perform logical operations in digital computers by representing true as 1 and false as 0. The fundamental logical operators are AND, OR, and NOT. Boolean algebra expressions can be represented in sum of products (SOP) form or product of sums (POS) form and minimized using algebraic rules or Karnaugh maps. Minterms and maxterms are used to derive Boolean functions from truth tables in canonical SOP or POS form.
Fixed-point and floating-point numbers can be represented in computers using binary numbers. Floating-point numbers represent numbers in scientific notation with a sign, mantissa, and exponent. In 8-bit floating point, numbers use 1 bit for sign, 3 bits for exponent, and 4 bits for mantissa, such as 0.001 x 21 = 2.25. Larger precision formats such as 32-bit and 64-bit floating point according to the IEEE standard use more bits for exponent and mantissa.
A combinational circuit is a logic circuit whose output is solely determined by the present input. It has no internal memory and its output depends only on the current inputs. A half adder is a basic combinational circuit that adds two single bits and produces a sum and carry output. A full adder adds three bits and produces a sum and carry like the half adder. Other combinational circuits discussed include half and full subtractors, decoders, encoders, and priority encoders.
Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
This document discusses binary subtraction and multiplication. It provides 4 rules for binary subtraction: 0-0=0, 0-1=1, 1-0=1, and 1-1=0. An example of binary subtraction is shown subtracting 10010 from 10110 with a difference of 00100. Binary multiplication is also covered, noting the multiplier is always 1 or 0, and the 4 basic rules are: 0x0=0, 0x1=0, 1x0=0, and 1x1=1. An example of binary multiplication is shown multiplying 10001111 by 1101000 to get a product of 1101.
Digital systems use logic gates like AND, OR, NOT, NAND, NOR, EXOR and EXNOR. The AND gate outputs 1 only if all inputs are 1. The OR gate outputs 1 if one or more inputs are 1. The NOT gate inverts the input. NAND and NOR gates are combinations of AND/OR gates with NOT gates. EXOR outputs 1 if either but not both inputs are 1, while EXNOR does the opposite. Truth tables define the output of each gate combination.
In electronics, an adder is a digital circuit that performs addition of numbers.
In modern computers and other kinds of processors, adders are used in the arithmetic logic unit (ALU), but also in other parts of the processor, where they are used to calculate addresses, table indices, and similar operations.
This document defines and classifies different types of binary codes. It explains that binary codes represent numeric and alphanumeric data as groups of bits. Binary codes are classified as weighted or non-weighted, reflective or non-reflective, and sequential or non-sequential. Common binary codes include ASCII, EBCDIC, Hollerith, BCD, excess-3, and Gray codes. Error detecting and correcting codes are also discussed which add extra bits to detect or correct errors during data transmission. Examples of different binary codes are provided.
This document provides an outline for a course on digital logic design. It includes the course title and credit hours, topics that will be covered such as Boolean algebra, logic gates, combinational and sequential circuits, programmable logic devices, and memory. It also lists recommended textbooks and provides the grading breakdown. Examples of analogue and digital quantities, signals, and number systems are given. Common logic gates such as AND, OR, NOT, NAND and NOR are described along with their truth tables and applications. Combinational circuits, functional devices, sequential circuits and memory are also introduced.
The document discusses binary subtraction and different types of binary subtractors. It describes half subtractors and full subtractors. A half subtractor is a basic circuit that can subtract two binary bits and outputs the difference and borrow. A full subtractor can subtract three bits by also considering the borrow from the previous stage. Truth tables and K-maps are used to derive the logic equations for difference and borrow outputs. Full subtractors are realized using basic gates by complementing one input to convert a full adder circuit into a full subtractor.
presentation on (Boolean rules & laws)kinza arshad
This document contains a presentation outline on Boolean algebra that was presented to a professor. The presentation covered the rules of Boolean algebra, laws of Boolean algebra, and examples of simplifying Boolean expressions. Some key points include:
- The rules of Boolean algebra include basic logical operations like A+1=1 and A.A=0.
- The laws include commutative, associative, and distributive laws that govern how logical operations are rearranged.
- Examples showed simplifying expressions using the rules and laws, like applying distribution to break apart expressions.
- Truth tables were included to verify the logical equivalencies hold for all possible variable values.
Binary arithmetic is essential for digital computers and systems. It involves adding, subtracting, multiplying, and dividing binary numbers using basic rules. Signed binary numbers represent positive and negative values using sign-magnitude, 1's complement, and 2's complement methods. Arithmetic operations on signed binary numbers follow rules for handling the sign bit and complement representations.
1) Karnaugh maps provide a systematic method for simplifying Boolean expressions and minimizing them to their simplest forms.
2) Karnaugh maps arrange variables in a two-dimensional grid where each cell represents a minterm and adjacent cells differ in only one variable.
3) Expressions can be minimized by grouping adjacent cells containing 1s and eliminating any variables that change across the group's boundaries.
Karnaugh maps (K-maps) are used to simplify Boolean logic expressions. A K-map arranges the minterms from a truth table into an array of cells where adjacent cells differ by only one variable. Groups of adjacent 1s in the K-map correspond to terms in a sum-of-products expression. The process of mapping a logic function onto a K-map and grouping 1s results in a minimum simplified expression. Don't care conditions can be treated as 1s to form larger groups for greater simplification. Both sum-of-products and product-of-sums expressions can be mapped and minimized using K-maps.
1. Boolean algebra is a mathematical system used to specify and transform logic functions. It uses binary variables that take on values of 1 or 0 and logical operators like AND, OR, and NOT.
2. Logic gates implement logic functions physically using electronic components. Common gates are AND, OR, and NOT. Gates have a small but nonzero delay between input change and output change.
3. Boolean expressions can be represented using truth tables, logic diagrams, or algebraic expressions. Standard forms include sum-of-minterms and product-of-maxterms forms.
The document discusses binary arithmetic operations including addition, subtraction, multiplication, and division. It provides examples and step-by-step explanations of how to perform each operation in binary. For addition and subtraction, it explains the rules and concepts like carry bits and two's complement. For multiplication, it describes the shift-and-add method. And for division, it outlines the long division approach of shift-and-subtract in binary.
Lecture 05-Logic expression and Boolean Algebra.pptxWilliamJosephat1
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.
This document provides an overview of Boolean algebra and logic gates. It introduces Boolean logic operations like AND, OR, and NOT. It covers Boolean algebra laws and De Morgan's theorems. It also discusses logic gate types like AND, OR, NOT, NAND, NOR, XOR and XNOR. Karnaugh maps are introduced as a method to simplify Boolean expressions.
This document provides information about Boolean algebra and logic gates. It defines Boolean algebra as a subarea of algebra dealing with variables that can have only two values, true or false. The basic Boolean operations are AND, OR, and NOT. Laws of Boolean algebra like commutative, associative, distributive, and De Morgan's theorems are explained and used to simplify Boolean expressions. Boolean expressions can be represented in sum of products or product of sums form. The document also introduces logic gates and how Boolean operators map to logic gates.
This document discusses simplification of Boolean algebra expressions using techniques like Boolean algebra laws and Karnaugh maps. It begins by presenting an example of simplifying an expression using distributive and other laws. It then discusses standard forms like sum-of-products (SOP) and product-of-sums (POS), and how to convert between forms. Truth tables are presented as a way to represent Boolean expressions. Finally, Karnaugh maps are introduced as a systematic method to minimize expressions into their simplest SOP or POS form.
The document discusses different digital logic components including logic gates, flip flops, registers, and counters. It describes the basic types of logic gates such as AND, OR, NOT, NAND, and NOR gates. It also discusses different types of flip flops including T, S-R, J-K, and D flip flops which are used to store binary data. Registers are formed using groups of flip flops to store multi-bit data. Counters are also discussed as another component of digital logic systems.
This document discusses Boolean algebra concepts including logic gates, Boolean expressions, truth tables, and logic circuit design. It covers:
1) The definitions and truth tables of common logic gates like AND, OR, NAND, and NOR.
2) How to derive the Boolean expression of a logic circuit and construct a circuit from a given Boolean expression.
3) Converting between Sum of Products (SOP) and Product of Sums (POS) forms using techniques like standard SOP from a truth table.
This document discusses Boolean algebra and logic gates. It defines Boolean algebra as a mathematical system using two values, typically true/false or 1/0. Boolean expressions are created using common operators like AND, OR, and NOT. Truth tables define the outputs of these operators. Logic gates are physical implementations of Boolean operators, including AND, OR, NAND, and NOR gates. Laws like De Morgan's theorem and the properties of universal gates like NAND and NOR are also covered.
George Boole developed Boolean algebra in 1854 to simplify and analyze complex logical expressions using binary logic. Boolean algebra uses logical symbols like 0 and 1 to represent digital circuit inputs and outputs, and laws and rules to reduce complex expressions to equivalent simpler forms using fewer logic gates. Some key laws and rules include the commutative, associative, distributive, absorption, and De Morgan's laws. Boolean functions describe logical relationships between variables and can be represented by algebraic expressions or truth tables. Methods like Karnaugh maps are used to minimize expressions.
The document discusses logic gates and Boolean algebra. It defines key logic gate terms like AND, OR, NAND, NOR, and XOR gates. It provides truth tables that define the output of each gate based on all possible input combinations. Boolean algebra laws and operations are also covered, including addition, multiplication, commutative laws, associative laws, and the distributive law. Methods for converting between Boolean expressions, truth tables, and logic circuits are described. Examples are provided to illustrate how to derive the expression, truth table, or circuit from one of the other representations.
This document provides an overview of digital circuits and logic concepts. It discusses number systems including binary, octal, hexadecimal and complements. It also covers logic gates, Boolean algebra, Karnaugh maps, logic families and integrated circuits. Specifically, it defines OR, AND, NOT and universal gates. It describes properties of Boolean algebra including commutative, associative and distributive properties. It also explains DeMorgan's theorems, sum of products and product of sums expressions. Finally, it discusses different logic families including transistor-transistor logic and metal-oxide semiconductor circuits.
- Boolean algebra uses binary numbers (0 and 1) and logical operations (AND, OR, NOT) to analyze and simplify digital circuits.
- It was invented by George Boole in 1854 and represents variables that can be either 1 or 0.
- The document discusses Boolean operations, laws, logic gates, minimization techniques, and representing functions as sums of products.
The document discusses various digital logic gates including inverters, AND gates, OR gates, NAND gates, NOR gates, XOR gates, and XNOR gates. It provides the truth tables and logic expressions for each gate. It also covers Boolean algebra concepts such as Boolean operations, laws, and theorems including DeMorgan's theorems. Venn diagrams are used to illustrate some Boolean algebra rules and theorems.
Boolean Aljabra.pptx of dld and computeritxminahil29
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George Boole first introduced Boolean algebra in 1854 as a way to systematically analyze logic circuits. Boolean algebra uses variables and operations like AND, OR and NOT to represent the behavior of digital logic gates. A key insight was Claude Shannon's 1938 application of Boolean algebra to the analysis and design of logic circuits. Boolean algebra provides a concise way to represent the operation of any logic circuit and determine its output for all combinations of inputs.
George Boole developed Boolean algebra between 1815-1864 as an algebra of logic to represent logical statements as algebraic expressions. Boolean algebra uses two values, True and False (represented by 1 and 0 respectively) and logical operators like AND, OR, and NOT to represent logical statements and perform operations on them. Boolean algebra finds application in digital circuits where it is used to perform logical operations. Canonical forms and Karnaugh maps are techniques used to simplify Boolean expressions into their minimal forms.
Boolean algebra deals with logical operations on binary variables that have two possible values, typically represented as 1 and 0. George Boole first introduced Boolean algebra in 1854. Boolean algebra uses logic gates like AND, OR, and NOT as basic building blocks. Positive logic represents 1 as high and 0 as low, while negative logic uses the opposite. Boolean algebra laws and Karnaugh maps are used to simplify logical expressions. Don't care conditions allow for groupings in K-maps that further reduce expressions.
George Boole first introduced Boolean algebra in 1854 as a way to systematically analyze logic circuits. Boolean algebra represents logical variables and operations using algebra, allowing the behavior of circuits to be expressed with equations. A Boolean expression can be derived for any logic circuit and a truth table constructed to show the output for all combinations of inputs.
This document discusses digital logic design and Boolean algebra. It covers topics like the rules of Boolean algebra, Demorgan's theorems, sum of product (SOP) form, and product of sum (POS) form. The introduction defines logic simplification and different simplification methods like Boolean algebra, Karnaugh maps, and Quine-McCluskey. It then provides details on the basic rules and laws of Boolean algebra, including commutative, associative, and distributive laws. Demorgan's theorems and examples of their applications are also presented. Methods for simplifying Boolean expressions using Boolean algebra rules are demonstrated. Finally, it discusses converting Boolean expressions to standard SOP and POS forms.
Here are the steps to solve this problem:
(a) Create a truth table with inputs A, B, C and output F
(b) Draw the logic circuit diagram using the given Boolean expression
(c) Apply Boolean algebra rules like absorption, idempotent, etc. to simplify the expression
(d) Create a new truth table using the simplified expression and check if it matches the original table
(e) Draw the new simplified logic circuit diagram and note the reduction in number of gates
This provides a systematic approach to solve Boolean algebra problems - creating truth tables, drawing circuit diagrams, simplifying expressions using rules, and verifying the results. Let me know if any part needs more explanation.
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### Advantages of TDM
- **Efficient Use of Bandwidth**: TDM all
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Boolen Algebra Lecture Notes.pdf
1. BOOLEAN ALGEBRA
Let X be a nonempty set with two binary operations + and ∗, a unary
operation ‘, and two distinct elements 0 and 1. Then X is called a Boolean
algebra if the following axioms hold where A, B, C are any elements in X:
Commutative Laws
A + B = B + A
A ∗ B = B ∗ A
Distributive Laws
A ∗ B + C = (A ∗ B) + (A ∗ C)
A + B ∗ C = A + B ∗ (A + C)
Complement Laws
A + A = 1
A ∗ A = 0
Identity Laws
A + 0 = A
A ∗ 1 = A
2. BOOLEAN ALGEBRA
The operations +, ∗, and ‘ are called sum, product, and complement,
respectively.
Boolean Algebra is the mathematics of digital systems.
Definition of terms
Variable- symbol used to represent a logical quantity. Any variable can
have a value of 1 or 0.
Complement- inverse of a variable. A or A
Literal is a variable/ complement of a variable.
Boolean addition is equivalent to the OR operation.
Basic rules for Boolean addition:
0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1
In Boolean algebra, a sum term is a sum of literals. e.g. A + B
3. BOOLEAN ALGEBRA
Basic rules useful in manipulating and simplifying Boolean expressions:
Idempotent laws:
𝐀 ∗ 𝐀 = 𝐀 𝐀 + 𝐀 = 𝐀
Boundedness laws:
𝐀 + 1 = 𝟏 𝐀 ∗ 𝟎 = 𝟎
Absorption laws:
𝐀 + 𝐀 ∗ 𝐁 = 𝐀 𝐀 ∗ (𝐀 + 𝐁) = 𝐀
Involution law: 𝐀 = 𝐀
𝐀 + A𝐁 = 𝐀 + 𝐁 𝐀 + 𝐁 𝐀 + 𝐂 = 𝐀 + 𝐁𝐂
Boolean multiplication is equivalent to the AND operation.
Basic rules for Boolean multiplication:
0 ∗ 0 = 0 0 ∗ 1 = 1 1 ∗ 0 = 1 1 ∗ 1 = 1
In Boolean algebra, a product term is a product of literals. e.g A ∗ B
4. BOOLEAN ALGEBRA
De Morgan Laws
The complement of a product of variables is equal to the sum of the
complements of the variables
XY = X + Y
The complement of a sum of variables is equal to the product of the
complements of the variables
X + Y = X. Y
5. BOOLEAN ALGEBRA
Simplification of Boolean expressions using Boolean Algebra
Its reducing a particular expression to its simplest form or change its form
to a more convenient one.
Use basic laws, rules and theorems of Boolean Algebra
Simplified Boolean expression uses the fewest gates possible to implement
a given expression.
Example:
Simplify the following Boolean Expression:
𝐀𝐁 + 𝐀𝐂 + 𝐀𝐁𝐂
= 𝐀𝐁 𝐀𝐂 + 𝐀𝐁𝐂 apply De Morgan law
= ( 𝐀 + 𝐁 )(𝐀 + 𝐂) + 𝐀𝐁𝐂 apply De Morgan law
= (𝐀 + 𝐁𝐂) + 𝐀𝐁𝐂 apply the rule 𝐀 + 𝐁 𝐀 + 𝐂 = 𝐀 + 𝐁𝐂
= 𝐀 𝟏 + 𝐁𝐂 + 𝐁𝐂 apply the rule 𝟏 + 𝐀 = 𝟏
= 𝐀 + 𝐁𝐂
6. BOOLEAN ALGEBRA
Truth Table
It is a table showing the inputs and the corresponding outputs of a logic
expression or circuit.
Steps taken when constructing a truth table
List the input variables combinations of 0s and 1s in a binary sequence
( 2n combinations for n inputs).
Place a 1 or 0 in the output column for each combination of input
variables that was determined in the evaluation.
Example
Construct a truth table and put the results of logic circuit A(B + CD)
4 variables means 24
combinations for n inputs.
8. BOOLEAN ALGEBRA
Logic Gates
A logic gate is an circuit that perform basic logical operation.
In reality, gates consist of one to six transistors, but digital designers
think of them as a single unit.
Integrated circuits contain collections of gates suited to a particular
purpose.
The three simplest gates:
AND
OR
NOT
10. BOOLEAN ALGEBRA
Not gate/ Inverter
performs inversion/ complementation operation- changes one logic level
to the opposite level
The negation indicator is a bubble.
11. BOOLEAN ALGEBRA
And Gate
Its composed of 2/ more inputs and a single output.
the total number of possible combinations of binary inputs is given
by N = 2n where N is the input combinations and n is the number of
input variables.
performs logical multiplication
produces a HIGH output when all of the inputs are HIGH
12. BOOLEAN ALGEBRA
Or Gate
Its composed of 2/ more inputs and a single output.
performs logical addition.
produces a HIGH output when any of the inputs is HIGH.
13. BOOLEAN ALGEBRA
NAND gate
Contraction of NOT-AND i.e. AND function with a complemented output.
produces a LOW output only when all of the inputs are HIGH; the output
will be HIGH when any of the inputs is LOW.
DeMorgan’s Theorem: AB = A + B
NOR gate
Contraction of NOT-OR i.e. OR function with a complemented output.
produces a LOW output when any of the inputs are HIGH; the output will
be HIGH when all of the inputs are LOW.
DeMorgan’s Theorem: A + B = AB
15. BOOLEAN ALGEBRA
Boolean algebra provides a concise way to express the operation of a logic
circuit formed by a combination of logic gates so that the output can be
determined for various combinations of input values.
To derive a Boolean expression for a given logic circuit, begin at the left-
most inputs and work towards the final output, writing the expression for
each gate.
Example:
𝐀
𝐀 + 𝐁
𝐀 𝐀 + 𝐁 = 𝐀𝐁
𝐀
𝐁
𝐀 + 𝐁 + 𝐀𝐁
16. BOOLEAN ALGEBRA
Standard Forms of Boolean Expressions
Standardization makes the evaluation, simplification and implementation of
Boolean expression much more systematic and easier.
Domain of Boolean expression is the set of variables contained in the
expression in complemented or un complemented form.
The domain of the expression 𝐀 + 𝐀𝐁𝐂 + 𝐁𝐂𝐃 is the set of variables
A, B, C ,D.
All Boolean expressions, regardless of their form can be converted into 2
standard forms:
Sum-of-Product
Product-of-Sum
17. BOOLEAN ALGEBRA
Sum of Products
Its formed when 2 or more product terms that are summed by Boolean
addition.
It can contain a single-variable term as in 𝐀 + 𝐀𝐁𝐂 + 𝐁𝐂𝐃
In SOP expression, a single overbar cannot extend over more than one
variable.
SOP expression can have 𝐀𝐁𝐂 but not 𝐀𝐁𝐂
Implementing a SOP expression simply ORing the outputs of 2 or more AND
gates( AND-OR implementation of SOP expression).
SOP can be implemented by NAND-NOR.
Conversion of a general expression to SOP form is done using Boolean
algebra techniques.
A standard SOP expression is one in which all the variables in the domain
appear in each product term in the expression
NB : Important in constructing truth table and in the K-map simplification
method
18. BOOLEAN ALGEBRA
To converting product terms to standard SOP
Multiply each nonstandard product term by A + A = 1 where A is the
missing variable.
Repeat the step above until all resulting product terms contain all
variables in the domain.
A standard product term = 1 for only one combination of variable values.
A product term is implemented with an AND gate whose output is 1 only if
each of its inputs is 1.
SOP expression is =1 only if one or more product terms in the expression =1.
ABCD = 1010 = 1
19. BOOLEAN ALGEBRA
Example:
Convert ABC + AB + ABCD into standard SOP form.
Standardize the first product term
ABC(D + D) = ABCD + ABCD
Standardize the second product term
AB (C + C) = ABC + ABC
ABC(D + D) = ABCD + ABCD
ABC (D + D) = ABCD + ABCD
Standard SOP
ABCD+ ABCD + ABCD+ ABCD + ABCD+ ABCD + ABCD
20. BOOLEAN ALGEBRA
Product of Sums
Formed when 2 or more sum terms multiplied.
A POS expression can contain a single-variable term.
In a POS expression, a single overbar cannot extend over more than
one variable.
Implementing a POS expression requires ANDing the outputs of 2 or
more OR gates.
A standard POS expression is one in which all variables in the domain
is in each sum term.
To convert sum terms to standard POS
Add to each non-standard product term AA = 0 where A is the
missing variable.
Apply rule A + BC = A + B A + C
Repeat step 1, until all resulting sum terms contain all variables in
the domain.
21. BOOLEAN ALGEBRA
A standard sum term =0 for only one combination of variable values and is 1
for all other combinations of values for variables.
A + B + C + D = 0 + 1 + 0 + 1 = 0
A sum term is implemented with OR gate whose output is 0 only if each of
its inputs =0.
A POS expression = 0 only if one or more of the sum in the expression =0.
22. BOOLEAN ALGEBRA
Standard SOP to Standard POS Conversion
Binary values of the product terms in a given standard SOP expression are
present in the equivalent standard POS expression.
To converting standard SOP to standard POS
Evaluate each product term in the SOP. Determine the numbers that
represent the product term.
REMEMBER: Standard product term =1
Determine all binary not included in the evaluation in 1
Write the equivalent sum term for each binary number from step 2 and
expression in POS form.
23. BOOLEAN ALGEBRA
Example
Convert the SOP expression to an equivalent POS expression:
ABC + ABC + ABC + ABC + ABC
Binary values are 000, 010, 011, 101,111
There are 3 variables in the domain of this expression and 23
possible
combinations.
SOP expression contains 5 of these combinations so the POS must contain the
other which are 001, 100 and 110.
A + B + C A + B + C (A + B + C)
To convert a standard POS to a standard SOP
Evaluate each sum term in the POS. Determine the numbers that
represent the sum term.
REMEMBER: Standard sum term =0
Determine all binary not included in the evaluation in 0
Write the equivalent product term for each binary number from step 2
24. BOOLEAN ALGEBRA
Steps considered when entering a SOP expression into the TRUTH Table
Convert the SOP expression standard form
List all possible combinations of binary values of the variables in the
variables in the expressions.
Place 1 in the output column for each binary value that makes the
standard SOP =1.
Place 0 for all the remaining binary values.
Example
Develop a truth table for the standard SOP expression ABC + ABC + ABC.
ABC: 001 ABC: 100 ABC: 111
25. BOOLEAN ALGEBRA
Inputs Output
A B C X Product Term
0 0 0 0
0 0 1 1 ABC
0 1 0 0
0 1 1 0
1 0 0 1 ABC
1 0 1 0
1 1 0 0
1 1 1 1 ABC
26. BOOLEAN ALGEBRA
Steps considered when entering a POS expression into the
TRUTH Table
Convert the POS expression standard form
List all possible combinations of binary values of the variables in the
variables in the expressions.
Place 0 in the output column for each binary value that makes the
standard POS=0.
Place 1 for all the remaining binary values.
Example
Develop a truth table for the standard POS expression 𝐀𝐁𝐂 + 𝐀𝐁𝐂 +
𝐀𝐁𝐂.
𝐀𝐁𝐂: 001 𝐀𝐁𝐂: 𝟏𝟎𝟎 𝐀𝐁𝐂: 111
27. BOOLEAN ALGEBRA
Karnaugh Map
K map provides a systematic method for simplifying Boolean expression and
if properly used will introduce the simplest SOP/ POS expression
possible(minimum expression).
Its an array of cells in which each cell represents a binary value of the input
variable.
Cells are arranged in a way so that simplification of a given expression is a
matter of grouping the cells.
K-map can be used for expressions with 2,3,4 and 5 variables.
The Quine McClusky method can be used for higher numbers of variables.
The number of cells in a K-map = total number of possible input variable
combinations.
28. BOOLEAN ALGEBRA
3 variable K-map array of 8 cells
Cells in a K-maps are arranged so that there is a single-variable change
between adjacent cells( adjacency is defined by a single-variable change).
Physically, each cell is adjacent to the cells that are immediately next to it on
any of its 4 sides( a cell is not adjacent to the cells to any of its corner –cells
that diagonally touch each other)
C
AB 0 1
00
01
11
10
ABC ABC
ABC ABC
ABC ABC
ABC ABC
29. BOOLEAN ALGEBRA
Top row cells are adjacent to the corresponding bottom row cells.
Cells in the outer left column are adjacent to the corresponding cells in the
outer right column( wrap- around adjacency).
K-map SOP Minimization
For an SOP expression in standard form, a 1 is placed on the K-map for each
product term in the expression.
The cells that do not have 1 are the cells for which the expression is 0.
When working with SOP expressions the 0s are left off the map.
30. BOOLEAN ALGEBRA
Example
Map the standard SOP expression on the K-map ABC + ABC + ABC +
ABC
For a nonstandard SOP expression , convert it to standard form before
you use a K-map.
C
AB 0 1
00
01
11
10
1
1
1 1
ABC
ABC
ABC
ABC
31. BOOLEAN ALGEBRA
Simplification of SOP expressions using K-maps is the process of
getting the fewest possible terms with the fewest possible
variables(minimum expression)
The minimum expression is obtained by grouping 1s i.e. enclosing
those adjacent cells containing 1s.
RULES
Groups must contain either 1,2,4,8 or 16 cells(2n cells).
Each cell in the group must be adjacent to 1 or more cells in that same
group.
Include the largest possible number of 1s in a group.
Each 1 on the map must be included in at least 1 group. The 1s
already in a group can be included in another group.
Groups may overlap.
Maximize the size the groups and minimize the number of groups.
32. BOOLEAN ALGEBRA
Determination of SOP expressions from K-map
Each group of cells containing 1s creates one product term composed of
variables that stay the same within a group . i.e. variables that do not change
from in complemented to uncomplemented or vice versa.
Determine the minimum product term for each group.
3 variable map
1 cell group yields a 3 variable term.
2 cell group yields a 2 variable term.
4 cell group yields a 1 variable term.
8 cell group yields a value of 1 for expression.
4 variable map
1 cell group yields a 4 variable term.
2 cell group yields a 3 variable term.
4 cell group yields a 2 variable term.
8 cell group yields a 1 variable term.
16 cell group yields a value of 1 for expression.
33. BOOLEAN ALGEBRA
Example
Minimize 𝐁𝐂𝐃 + 𝐀𝐁𝐂𝐃 + 𝐀𝐁𝐂𝐃 + 𝐀𝐁𝐂𝐃 + 𝐀𝐁𝐂𝐃 + 𝐀𝐁𝐂𝐃 +
𝐀𝐁𝐂𝐃 + 𝐀𝐁𝐂𝐃 + 𝐀𝐁C𝐃 using a K-map.
The first product term is not in standard form. Converting it into
standard form it becomes A𝐁𝐂 𝐃 + 𝐀𝐁𝐂 𝐃
A group of 8 cells
formed by adjacent
outer column produces
𝐃
A group of 4 cells
formed by adjacent
outer column produces
𝐁𝐂
Minimum expression 𝐁𝐂 + 𝐃
CD
AB 00 01 11 10
00
01
11
10
1 1 1
1 1
1 1
1 1 1
34. BOOLEAN ALGEBRA
With POS expressions in standard form, 0s representing the standard terms
are on the K-map.
Example
Map the standard POS expression on the K-map
(𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 + 𝐂 + 𝐃)
For a nonstandard POS expression , convert it to standard form before you
use a K-map.
K-maps POS Minimization
1. Determine the binary value of each sum term in the standard POS
expression with binary value =0.
2. Group the 0s to minimum sum terms
Example
Map the standard POS expression on the K-map
(𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 +
𝐂 + 𝐃)
35. BOOLEAN ALGEBRA
Conversion between POS and SOP using the K-map
With POS expressions, all cells that do not contain 0s contain 1s from which
SOP expression is derived.
Likewise for an SOP expression, all the cells that do not contain 1s contain
0s, from which the POS expression is derived.
This provides a good way to compare both minimum forms of an expression
to determine if one of them can be implemented with fewer gates than the
other.
Example
Convert the following standard POS expression into a minimum POS
expression, a standard SOP expression and a minimum SOP expression
(𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 +
𝐁 + 𝐂 + 𝐃) (𝐀 + 𝐁 + 𝐂 + 𝐃)(𝐀 + 𝐁 + +𝐂 + 𝐃)