The document provides an introduction to Boolean algebra through the use of truth tables. It defines basic Boolean operations such as negation, conjunction, and disjunction. Examples are given to illustrate truth tables and how they can demonstrate logical equivalence between expressions. The rules of Boolean algebra are also presented, which can be used to simplify Boolean expressions without using truth tables. Exercises with solutions are provided to allow students to practice applying these concepts.
(8) Inquiry Lab - Proofs About the Pythagorean Theoremwzuri
The document describes how to prove the Pythagorean theorem and its converse using geometric constructions and algebra. To prove the Pythagorean theorem, right triangles are arranged to form squares whose areas represent the expressions a2, b2, and c2. Equating the areas shows that a2 + b2 = c2. To prove the converse, it is shown that if a triangle satisfies a2 + b2 = c2, then it must be a right triangle by constructing another right triangle with the same properties. A two-column proof formally proves the converse statement.
This document is from a textbook chapter on Karnaugh maps. It provides an overview of how Karnaugh maps can be used to systematically simplify switching functions with up to five variables. Key points covered include:
- How to construct Karnaugh maps for 2, 3, 4, and 5 variable functions from truth tables or minterm/maxterm expressions.
- How adjacent terms on the map that differ in one variable can be combined using Boolean algebra rules.
- Examples showing how to simplify functions by combining adjacent 1s on the map to find minimum sum-of-products or product-of-sums expressions.
- Extensions like don't care terms and determining the minimum product-of-sums
This document summarizes a chapter on Boolean algebra from a 2004 textbook. It includes objectives, study guides, sections on multiplying and factoring expressions, exclusive-OR and equivalence operations, the consensus theorem, and algebraic simplification of switching expressions. The chapter provides examples and exercises to illustrate different theorems and methods for manipulating Boolean expressions algebraically.
This document discusses query optimization techniques in database systems. It provides examples of equivalence rules that can be used to optimize queries by rewriting them in equivalent forms that may be more efficient to evaluate. It also discusses strategies for efficiently evaluating join queries, including the use of indexes. Some key points made include:
- Equivalence rules allow rewriting queries in forms that avoid unnecessary intermediate results or that push selections past aggregation.
- The size of join results can be estimated based on cardinalities and selectivity of joins.
- Indexes on join attributes can improve join performance when used to lookup matching tuples.
This document introduces vectors and their properties. It defines vectors as quantities that have both magnitude and direction, unlike scalars which only have magnitude. The key concepts covered are:
- Representing vectors geometrically using directed line segments.
- Adding and multiplying vectors, including multiplying a vector by a scalar.
- Using position vectors to represent the location of points relative to a fixed origin.
- Expressing vectors in terms of other known vectors using properties of vector addition and scalar multiplication.
This document discusses properties of parallelograms. It provides three examples that demonstrate using properties of parallelograms to find missing angle measures, side lengths, and midpoints of diagonals. It also includes guided practice problems asking students to apply these properties. The key properties covered are that opposite sides of parallelograms are equal, opposite angles are equal, consecutive angles are supplementary, and diagonals bisect each other.
This document summarizes key concepts about proving segment relationships using postulates and theorems of geometry. It includes:
1) An introduction describing essential questions about segment addition and congruence proofs, as well as relevant postulates and theorems.
2) An example proof that if two segments are congruent, their endpoints are also congruent.
3) A proof of the transitive property of segment congruence.
4) An example proof using the properties of congruence and equality to show that if corresponding sides of a badge are congruent or equal, then the bottom and left sides are also congruent.
The document discusses Karnaugh maps, which are a graphical method for simplifying Boolean expressions.
[1] Karnaugh maps allow adjacent product terms in an expression to be grouped together and simplified using the rule that AB + AB = A. This simplification can reduce the complexity of the corresponding digital logic circuit.
[2] For an expression with two variables, the Karnaugh map is a 2x2 grid where each cell represents a product term. Adjacent cells that differ in only one variable can be grouped.
[3] For expressions with more variables, the Karnaugh map expands appropriately, with the edges wrapping like a cylinder. Worked examples show how maps can be
(8) Inquiry Lab - Proofs About the Pythagorean Theoremwzuri
The document describes how to prove the Pythagorean theorem and its converse using geometric constructions and algebra. To prove the Pythagorean theorem, right triangles are arranged to form squares whose areas represent the expressions a2, b2, and c2. Equating the areas shows that a2 + b2 = c2. To prove the converse, it is shown that if a triangle satisfies a2 + b2 = c2, then it must be a right triangle by constructing another right triangle with the same properties. A two-column proof formally proves the converse statement.
This document is from a textbook chapter on Karnaugh maps. It provides an overview of how Karnaugh maps can be used to systematically simplify switching functions with up to five variables. Key points covered include:
- How to construct Karnaugh maps for 2, 3, 4, and 5 variable functions from truth tables or minterm/maxterm expressions.
- How adjacent terms on the map that differ in one variable can be combined using Boolean algebra rules.
- Examples showing how to simplify functions by combining adjacent 1s on the map to find minimum sum-of-products or product-of-sums expressions.
- Extensions like don't care terms and determining the minimum product-of-sums
This document summarizes a chapter on Boolean algebra from a 2004 textbook. It includes objectives, study guides, sections on multiplying and factoring expressions, exclusive-OR and equivalence operations, the consensus theorem, and algebraic simplification of switching expressions. The chapter provides examples and exercises to illustrate different theorems and methods for manipulating Boolean expressions algebraically.
This document discusses query optimization techniques in database systems. It provides examples of equivalence rules that can be used to optimize queries by rewriting them in equivalent forms that may be more efficient to evaluate. It also discusses strategies for efficiently evaluating join queries, including the use of indexes. Some key points made include:
- Equivalence rules allow rewriting queries in forms that avoid unnecessary intermediate results or that push selections past aggregation.
- The size of join results can be estimated based on cardinalities and selectivity of joins.
- Indexes on join attributes can improve join performance when used to lookup matching tuples.
This document introduces vectors and their properties. It defines vectors as quantities that have both magnitude and direction, unlike scalars which only have magnitude. The key concepts covered are:
- Representing vectors geometrically using directed line segments.
- Adding and multiplying vectors, including multiplying a vector by a scalar.
- Using position vectors to represent the location of points relative to a fixed origin.
- Expressing vectors in terms of other known vectors using properties of vector addition and scalar multiplication.
This document discusses properties of parallelograms. It provides three examples that demonstrate using properties of parallelograms to find missing angle measures, side lengths, and midpoints of diagonals. It also includes guided practice problems asking students to apply these properties. The key properties covered are that opposite sides of parallelograms are equal, opposite angles are equal, consecutive angles are supplementary, and diagonals bisect each other.
This document summarizes key concepts about proving segment relationships using postulates and theorems of geometry. It includes:
1) An introduction describing essential questions about segment addition and congruence proofs, as well as relevant postulates and theorems.
2) An example proof that if two segments are congruent, their endpoints are also congruent.
3) A proof of the transitive property of segment congruence.
4) An example proof using the properties of congruence and equality to show that if corresponding sides of a badge are congruent or equal, then the bottom and left sides are also congruent.
The document discusses Karnaugh maps, which are a graphical method for simplifying Boolean expressions.
[1] Karnaugh maps allow adjacent product terms in an expression to be grouped together and simplified using the rule that AB + AB = A. This simplification can reduce the complexity of the corresponding digital logic circuit.
[2] For an expression with two variables, the Karnaugh map is a 2x2 grid where each cell represents a product term. Adjacent cells that differ in only one variable can be grouped.
[3] For expressions with more variables, the Karnaugh map expands appropriately, with the edges wrapping like a cylinder. Worked examples show how maps can be
The document defines different types of triangles based on their sides and angles. It discusses triangles formed by three non-collinear points connected by line segments. The types of triangles include scalene, isosceles, equilateral, acute, right, and obtuse triangles. Congruence rules for triangles are provided, including SAS, ASA, AAS, SSS, and RHS. Properties of triangles like angles opposite equal sides being equal and sides opposite equal angles being equal are explained. Inequalities relating sides and angles of triangles are described.
This document defines and discusses congruent triangles. It defines congruent as two figures with the same size and shape, and congruent polygons as polygons where all parts of one are congruent to matching parts of the other. Corresponding parts are the matching parts between congruent polygons that have the same position. The document provides examples of proving triangles congruent by identifying corresponding congruent parts and using triangle congruence theorems like CPCTC and the third angle theorem.
The document discusses Boolean algebra and logic simplification. It covers Boolean operations like addition, multiplication, and their relationships to logical OR and AND operations. Some key concepts covered include:
- Variables, complements, literals, sum terms, and product terms in Boolean algebra and their mappings to logic gates.
- Basic laws of Boolean algebra like commutative, associative, and distributive laws, which are analogous to ordinary algebra.
- Rules for simplifying Boolean expressions using logic gate equivalencies, like A+0=A, A+1=1, A.0=0, and DeMorgan's theorems.
- Applications of these laws, rules, and theorems to manipulate and
8.3 show that a quadrilateral is a parallelogramdetwilerr
This document discusses ways to show that a quadrilateral is a parallelogram using various theorems and properties of parallelograms. It provides examples of showing quadrilateral ABCD is a parallelogram by proving opposite sides are congruent and parallel using the distance and slope formulas. It also gives an example of using the fact that diagonals of a parallelogram bisect each other to determine the value of x that would make quadrilateral MNPQ a parallelogram. The document contains guided practice problems asking how to show a given quadrilateral is a parallelogram and for what value of x another would be a parallelogram.
This document provides an unsolved sample test paper for mathematics with 4 sections:
Section A contains 8 multiple choice questions worth 1 mark each. Section B contains 6 questions worth 2 marks each. Section C contains 10 questions worth 3 marks each involving calculations and proofs. Section D contains the most challenging questions, with 10 worth 4 marks each involving graphing, ratios, and geometric constructions. The test is out of a total of 90 marks and takes 3 hours to complete.
This document defines key terms related to congruent triangles such as congruent, congruent polygons, and corresponding parts. It also defines the Third Angle Theorem. Examples are provided to demonstrate how to prove triangles are congruent by identifying corresponding congruent parts and writing congruence statements. The document also includes multi-step proofs involving congruent triangles.
Rbse solutions for class 10 maths chapter 10 locus ex 10.1Arvind Saini
The document contains solutions to 8 questions about locus and geometry. The key points are:
(1) The locus of points equidistant from two lines is the line perpendicular to both lines.
(2) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
(3) The locus of points equidistant from three non-collinear points is a circle centered at their intersection point.
4.8 congruence transformations and coordinate geometrydetwilerr
1) The document discusses transformations in coordinate geometry including translations, reflections, and rotations. It provides examples of naming the type of transformation based on images and applying transformations to graphs.
2) Examples show applying coordinate rules for translations and reflections to graphs of figures and verifying that the resulting images are congruent.
3) Practice problems at the end review applying transformations like translation notation, reflecting figures, and identifying if a graph is a rotation of another and specifying the angle and direction.
This document summarizes key concepts from Chapter 1 of a textbook on Euclidean geometry. It begins by defining Pythagoras' theorem and its converse, providing proofs. It then discusses applications of Pythagoras' theorem, including Euclid's original proof. Later sections cover topics like constructing regular polygons, the cosine formula, Stewart's theorem, and Apollonius' theorem. Exercises provide additional problems relating to these geometric concepts.
The document contains a solved math exam paper for class 9 with 34 questions. It provides instructions that all questions are compulsory and carry varying marks. The questions cover a range of math topics like geometry, algebra, data representation and interpretation. Sample questions include finding the angle measures in geometric shapes, solving linear equations, constructing frequency distribution tables from data sets, calculating volumes of geometric solids, and properties of parallelograms, circles and triangles. The document also provides solutions for some of the questions as examples.
The document discusses coordinate proofs involving quadrilaterals. It provides examples of determining if quadrilaterals are congruent or similar by analyzing corresponding sides and angles. It also covers properties of parallelograms, rectangles, and rhombuses, such as having opposite sides that are congruent and opposite angles that are congruent. Examples are given of supplying missing coordinates to complete parallelograms and rectangles.
18 pc09 1.2_ digital logic gates _ boolean algebra_basic theoremsarunachalamr16
Digital logic gates are basic building blocks of digital circuits that make logical decisions based on input combinations. There are three basic logic gates: OR, AND, and NOT. Other common gates such as NAND, NOR, XOR, and XNOR are derived from these. Boolean algebra uses variables that can be 1 or 0, and logical operators like AND, OR, and NOT to represent logic functions. Logic functions can be expressed in canonical forms such as sum of minterms or product of maxterms. Standard forms like SOP and POS are also used. Conversions between these forms allow simplifying logic functions.
1. The document provides examples and explanations for determining if a quadrilateral is a rectangle, rhombus, or square based on properties of its angles and sides. Key theorems relate special properties of parallelograms to specific shapes.
2. Examples demonstrate using the slope and length of diagonals to classify quadrilaterals in the coordinate plane as rectangles, rhombuses, or squares. Additional information like a quadrilateral being a parallelogram is needed to apply the theorems.
3. A lesson quiz tests understanding of classifying quadrilaterals and determining validity of conclusions based on given information.
This was our written report in Modern Geometry :) I hope that this will be helpful to other students since we had a very difficult time in searching for references.
- The objectives of the lesson are to: add and subtract vectors using line segments, identify coplanar and parallel vectors using scalar multiples, and express coplanar and parallel vectors as a ratio.
- Key concepts covered include vector addition and subtraction, scalar multiples, ratios of vectors, and identifying parallel vectors.
- Examples are provided to demonstrate how to write vectors in terms of variables, find ratios of vectors, and express vectors in terms of given variables using properties of parallel vectors and scalar multiples.
This document discusses harmonic sets in projective geometry and their relationship to harmonics in music. It defines harmonic sets as sets of points on a line that divide the line harmonically. Harmonic sets are preserved under projectivities and collineations. They relate to the harmonic sequence and harmonic mean. In music, harmonic sets correspond to just musical intervals formed by ratios of small whole numbers, like octaves, perfect fifths and major thirds. An android app was created by Stephen Brown to demonstrate harmonic sets.
This module covers similarity and the Pythagorean theorem as they relate to right triangles. It discusses how the altitude to the hypotenuse of a right triangle divides it into two smaller right triangles that are similar to each other and the original triangle. It also explains how the altitude is the geometric mean of the hypotenuse segments. Special right triangles like 45-45-90 and 30-60-90 triangles are examined, relating side lengths through their properties. The Pythagorean theorem is derived and used to solve for missing sides of right triangles. Students work through examples and multi-step problems applying these concepts.
8.5 use properties of trapezoids and kitesdetwilerr
This document provides examples and explanations for using properties of trapezoids and kites to solve geometry problems. It includes 4 examples of finding missing angle measures in trapezoids and kites using their properties. It also provides 5 practice problems for students to work through, with explanations of the reasoning and steps to find lengths and angle measures in various trapezoids and kites. The document demonstrates how to apply theorems about parallel lines, congruent angles, and midsegments of trapezoids to solve for unknown values in multi-step word problems involving these special quadrilaterals.
This document provides information about congruence of triangles from a geometry textbook. It includes definitions of congruent figures and associating real numbers with lengths and angles. It describes the one-to-one correspondence test for congruence of triangles. It discusses sufficient conditions for congruence including SAS, SSS, ASA, and SAA. It presents activities and examples verifying these tests and exploring properties of isosceles and equilateral triangles. The document encourages critical thinking through "Think it Over" prompts and upgrading the chapter with additional content.
This document discusses properties of parallelograms. It begins with objectives and vocabulary, then provides examples of using properties of parallelograms to find missing measures. These properties include opposite sides being parallel, opposite angles being supplementary, and diagonals bisecting each other. The document also covers using parallelograms in coordinate proofs and two-column geometric proofs.
This document defines and provides examples of various poetic devices and forms of poetry. It discusses devices such as speaker, diction, imagery, allusion, simile, personification, metaphor, refrain, symbol, and stanza. It also provides examples of poetic forms like blackout poetry, haiku, sonnets, concrete poems, acrostic poems, free verse, parody poems, and odes. The examples illustrate how these devices and forms can be used in poetry.
This document summarizes research on the state of the electrical/power engineering industry from a survey of 476 professionals. It finds that most respondents work at consulting engineering firms and specify products for new commercial construction projects. Trends in the industry include a shift toward retrofitting and renovating existing buildings. Standards and codes are changing frequently, and interoperability of systems poses challenges for engineers. Mobile tools are increasingly important for accessing product information.
The document defines different types of triangles based on their sides and angles. It discusses triangles formed by three non-collinear points connected by line segments. The types of triangles include scalene, isosceles, equilateral, acute, right, and obtuse triangles. Congruence rules for triangles are provided, including SAS, ASA, AAS, SSS, and RHS. Properties of triangles like angles opposite equal sides being equal and sides opposite equal angles being equal are explained. Inequalities relating sides and angles of triangles are described.
This document defines and discusses congruent triangles. It defines congruent as two figures with the same size and shape, and congruent polygons as polygons where all parts of one are congruent to matching parts of the other. Corresponding parts are the matching parts between congruent polygons that have the same position. The document provides examples of proving triangles congruent by identifying corresponding congruent parts and using triangle congruence theorems like CPCTC and the third angle theorem.
The document discusses Boolean algebra and logic simplification. It covers Boolean operations like addition, multiplication, and their relationships to logical OR and AND operations. Some key concepts covered include:
- Variables, complements, literals, sum terms, and product terms in Boolean algebra and their mappings to logic gates.
- Basic laws of Boolean algebra like commutative, associative, and distributive laws, which are analogous to ordinary algebra.
- Rules for simplifying Boolean expressions using logic gate equivalencies, like A+0=A, A+1=1, A.0=0, and DeMorgan's theorems.
- Applications of these laws, rules, and theorems to manipulate and
8.3 show that a quadrilateral is a parallelogramdetwilerr
This document discusses ways to show that a quadrilateral is a parallelogram using various theorems and properties of parallelograms. It provides examples of showing quadrilateral ABCD is a parallelogram by proving opposite sides are congruent and parallel using the distance and slope formulas. It also gives an example of using the fact that diagonals of a parallelogram bisect each other to determine the value of x that would make quadrilateral MNPQ a parallelogram. The document contains guided practice problems asking how to show a given quadrilateral is a parallelogram and for what value of x another would be a parallelogram.
This document provides an unsolved sample test paper for mathematics with 4 sections:
Section A contains 8 multiple choice questions worth 1 mark each. Section B contains 6 questions worth 2 marks each. Section C contains 10 questions worth 3 marks each involving calculations and proofs. Section D contains the most challenging questions, with 10 worth 4 marks each involving graphing, ratios, and geometric constructions. The test is out of a total of 90 marks and takes 3 hours to complete.
This document defines key terms related to congruent triangles such as congruent, congruent polygons, and corresponding parts. It also defines the Third Angle Theorem. Examples are provided to demonstrate how to prove triangles are congruent by identifying corresponding congruent parts and writing congruence statements. The document also includes multi-step proofs involving congruent triangles.
Rbse solutions for class 10 maths chapter 10 locus ex 10.1Arvind Saini
The document contains solutions to 8 questions about locus and geometry. The key points are:
(1) The locus of points equidistant from two lines is the line perpendicular to both lines.
(2) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
(3) The locus of points equidistant from three non-collinear points is a circle centered at their intersection point.
4.8 congruence transformations and coordinate geometrydetwilerr
1) The document discusses transformations in coordinate geometry including translations, reflections, and rotations. It provides examples of naming the type of transformation based on images and applying transformations to graphs.
2) Examples show applying coordinate rules for translations and reflections to graphs of figures and verifying that the resulting images are congruent.
3) Practice problems at the end review applying transformations like translation notation, reflecting figures, and identifying if a graph is a rotation of another and specifying the angle and direction.
This document summarizes key concepts from Chapter 1 of a textbook on Euclidean geometry. It begins by defining Pythagoras' theorem and its converse, providing proofs. It then discusses applications of Pythagoras' theorem, including Euclid's original proof. Later sections cover topics like constructing regular polygons, the cosine formula, Stewart's theorem, and Apollonius' theorem. Exercises provide additional problems relating to these geometric concepts.
The document contains a solved math exam paper for class 9 with 34 questions. It provides instructions that all questions are compulsory and carry varying marks. The questions cover a range of math topics like geometry, algebra, data representation and interpretation. Sample questions include finding the angle measures in geometric shapes, solving linear equations, constructing frequency distribution tables from data sets, calculating volumes of geometric solids, and properties of parallelograms, circles and triangles. The document also provides solutions for some of the questions as examples.
The document discusses coordinate proofs involving quadrilaterals. It provides examples of determining if quadrilaterals are congruent or similar by analyzing corresponding sides and angles. It also covers properties of parallelograms, rectangles, and rhombuses, such as having opposite sides that are congruent and opposite angles that are congruent. Examples are given of supplying missing coordinates to complete parallelograms and rectangles.
18 pc09 1.2_ digital logic gates _ boolean algebra_basic theoremsarunachalamr16
Digital logic gates are basic building blocks of digital circuits that make logical decisions based on input combinations. There are three basic logic gates: OR, AND, and NOT. Other common gates such as NAND, NOR, XOR, and XNOR are derived from these. Boolean algebra uses variables that can be 1 or 0, and logical operators like AND, OR, and NOT to represent logic functions. Logic functions can be expressed in canonical forms such as sum of minterms or product of maxterms. Standard forms like SOP and POS are also used. Conversions between these forms allow simplifying logic functions.
1. The document provides examples and explanations for determining if a quadrilateral is a rectangle, rhombus, or square based on properties of its angles and sides. Key theorems relate special properties of parallelograms to specific shapes.
2. Examples demonstrate using the slope and length of diagonals to classify quadrilaterals in the coordinate plane as rectangles, rhombuses, or squares. Additional information like a quadrilateral being a parallelogram is needed to apply the theorems.
3. A lesson quiz tests understanding of classifying quadrilaterals and determining validity of conclusions based on given information.
This was our written report in Modern Geometry :) I hope that this will be helpful to other students since we had a very difficult time in searching for references.
- The objectives of the lesson are to: add and subtract vectors using line segments, identify coplanar and parallel vectors using scalar multiples, and express coplanar and parallel vectors as a ratio.
- Key concepts covered include vector addition and subtraction, scalar multiples, ratios of vectors, and identifying parallel vectors.
- Examples are provided to demonstrate how to write vectors in terms of variables, find ratios of vectors, and express vectors in terms of given variables using properties of parallel vectors and scalar multiples.
This document discusses harmonic sets in projective geometry and their relationship to harmonics in music. It defines harmonic sets as sets of points on a line that divide the line harmonically. Harmonic sets are preserved under projectivities and collineations. They relate to the harmonic sequence and harmonic mean. In music, harmonic sets correspond to just musical intervals formed by ratios of small whole numbers, like octaves, perfect fifths and major thirds. An android app was created by Stephen Brown to demonstrate harmonic sets.
This module covers similarity and the Pythagorean theorem as they relate to right triangles. It discusses how the altitude to the hypotenuse of a right triangle divides it into two smaller right triangles that are similar to each other and the original triangle. It also explains how the altitude is the geometric mean of the hypotenuse segments. Special right triangles like 45-45-90 and 30-60-90 triangles are examined, relating side lengths through their properties. The Pythagorean theorem is derived and used to solve for missing sides of right triangles. Students work through examples and multi-step problems applying these concepts.
8.5 use properties of trapezoids and kitesdetwilerr
This document provides examples and explanations for using properties of trapezoids and kites to solve geometry problems. It includes 4 examples of finding missing angle measures in trapezoids and kites using their properties. It also provides 5 practice problems for students to work through, with explanations of the reasoning and steps to find lengths and angle measures in various trapezoids and kites. The document demonstrates how to apply theorems about parallel lines, congruent angles, and midsegments of trapezoids to solve for unknown values in multi-step word problems involving these special quadrilaterals.
This document provides information about congruence of triangles from a geometry textbook. It includes definitions of congruent figures and associating real numbers with lengths and angles. It describes the one-to-one correspondence test for congruence of triangles. It discusses sufficient conditions for congruence including SAS, SSS, ASA, and SAA. It presents activities and examples verifying these tests and exploring properties of isosceles and equilateral triangles. The document encourages critical thinking through "Think it Over" prompts and upgrading the chapter with additional content.
This document discusses properties of parallelograms. It begins with objectives and vocabulary, then provides examples of using properties of parallelograms to find missing measures. These properties include opposite sides being parallel, opposite angles being supplementary, and diagonals bisecting each other. The document also covers using parallelograms in coordinate proofs and two-column geometric proofs.
This document defines and provides examples of various poetic devices and forms of poetry. It discusses devices such as speaker, diction, imagery, allusion, simile, personification, metaphor, refrain, symbol, and stanza. It also provides examples of poetic forms like blackout poetry, haiku, sonnets, concrete poems, acrostic poems, free verse, parody poems, and odes. The examples illustrate how these devices and forms can be used in poetry.
This document summarizes research on the state of the electrical/power engineering industry from a survey of 476 professionals. It finds that most respondents work at consulting engineering firms and specify products for new commercial construction projects. Trends in the industry include a shift toward retrofitting and renovating existing buildings. Standards and codes are changing frequently, and interoperability of systems poses challenges for engineers. Mobile tools are increasingly important for accessing product information.
This document discusses the relationship between marketing and engineering. It notes that marketing is too important to leave just to the marketing department and that marketing is a contest for people's attention. It suggests that engineers should educate marketers on technical jargon when presenting concepts and ideas to focus on the overall concept rather than just technical details.
Raising brand awareness and how to make the most of your communicationsIntelligentInk
Get some tangible tips for improving your organisation's profile and for creating engaging communications that will impress those who already support you, and attract those who you would like to support you - even if your budget is smaller than you would like!
Particularly great for not for profit organisations, this slideshow can benefit any business or organisation that need a helping hand when it comes to marketing, PR, and communications.
Health care facilities and hospitals have more stringent circuit protection requirements than conventional building electrical systems. Because of the unique constraints of health care facility electrical systems, design engineers must ensure these demanding requirements are met. However, codes adopted in many jurisdictions don’t provide a significant amount of guidance regarding feeder and branch circuit design for health care facilities. Engineers must understand and apply circuit protection best practices especially when designing electrical systems in health care facilities.
This document summarizes a presentation on transfer switches and switchgear for emergency power systems. It discusses applicable codes from NFPA 70, 110, and 99 and requirements for transfer switches regarding preventing interconnection, being listed for emergency use, and including signaling and time delays. It also covers open and closed transition switches, applying switches in designs, and using switch timing to reduce generator size through load sequencing.
New chiller requirements go into effect on Jan. 1, 2015. ASHRAE Standard 90.1-2010: Energy Standard for Buildings Except Low-Rise Residential Buildings, Addendum ch details minimum performance requirements of heating and air conditioning equipment, including chillers, boilers, and packaged equipment, which continue to increase from the previous standard.
Equipment efficiencies are increased for heat pumps, packaged terminal air conditioners, single package vertical heat pumps, air conditioners, and evaporative condensers. Additional provisions address commercial refrigeration equipment, improved controls on heat rejection and boiler equipment, requirements for expanded use of energy recovery, small motor efficiencies, and fan power control and credits. Control revision requirements have been added to the standard such as direct digital controls in many applications. Finally, the 2013 edition completes the work that was begun on equipment efficiencies for chillers in the 2010 edition.
While Addendum ch may simplify these requirements for consulting engineers, chiller manufacturers are faced with equipment redesigns to comply by the deadline. Also, designers must pay special attention to which path of compliance will be used when testing chillers with or without variable frequency drives (VFDs).
This document provides an overview of Boolean algebra. It begins by listing the key objectives of learning Boolean algebra, which include understanding AND, OR, and NOT logic gates, simplifying Boolean expressions, and minimizing circuits. The document then provides examples of logic gates like AND, OR, and NOT gates. It explains how circuits can be represented using Boolean expressions and switching tables. Several laws of Boolean algebra are defined, like distributive, commutative, and De Morgan's laws. The concept of Boolean functions is introduced, where a function defines the relationship between inputs and outputs. Truth tables are used to represent both Boolean functions and to derive a function from a given truth table.
This document provides an overview of key concepts in sets and logic covered in Chapter 1, including:
- Basic symbols and terminology used in sets and logic
- Definitions of sets, set operations like union and intersection, and relationships between sets
- Properties of sets such as subsets, power sets, cardinality, and complement
- Algebraic laws governing set operations like distributive, associative, commutative, identity, and De Morgan's laws
- Methods of representing and reasoning about sets using mathematical notation and Venn diagrams
Pythagorean Theorem and its various ProofsSamanyou Garg
The document discusses several proofs of the Pythagorean theorem provided by different mathematicians. It begins by stating the theorem, then provides 6 different proofs: the first given by President James Garfield in 1876 using a trapezoid approach; the second using similarity of triangles; the third constructing a square from 4 copies of a right triangle; the fourth also using similarity; the fifth by rearrangement of the formula; and the sixth using a geometric representation of the areas. It also discusses some applications of the theorem in fields like architecture, navigation, and coordinate geometry.
This document discusses Boolean algebra and its application to sets and switching circuits. It begins by defining a Boolean algebra as a set with binary operations of sum and product satisfying certain properties like closure, commutativity, associativity, distributivity, identities, and complements. It then provides examples of Boolean algebras including the set {1,0} and sets closed under union and intersection. The document proceeds to cover topics like duality, basic theorems involving idempotent laws and De Morgan's laws, order relations, and an application to switching circuit design where circuits can be modeled as Boolean expressions.
The document discusses set theory and relations. It defines sets and subsets, set operations including union, intersection, and complement, and properties of sets like cardinality and power sets. Examples are provided to demonstrate counting elements in sets and using Venn diagrams to represent relationships between sets.
This document provides an overview of set theory concepts including:
- Sets, elements, and set operations like union, intersection, difference, and complement.
- Finite and countable sets versus infinite sets.
- Product sets involving ordered pairs from two sets.
- Classes of sets including the power set of a set, which contains all subsets.
The document defines a Boolean algebra as a set with binary operations of sum and product that satisfy closure, commutative, associative, distributive, identity, and complement laws. Examples of Boolean algebras include sets of binary values {1,0} and sets closed under union, intersection, and complement. Theorems proved include idempotent, involution, De Morgan's, and order properties. Boolean switching circuits can be described using series and parallel combinations of switches and satisfy the algebra of a Boolean algebra.
This document discusses concepts related to statics including:
1. It covers four main topics - the condition of equilibrium of coplanar concurrent forces, the concept of a free body diagram, the sine rule for triangles, and Lami's theorem.
2. Lami's theorem states that if three coplanar forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces.
3. Several examples are provided to demonstrate how to apply the sine rule and Lami's theorem to calculate tensions in strings and magnitudes of forces.
Three Solutions of the LLP Limiting Case of the Problem of Apollonius via Geo...James Smith
This document adds to the collection of solved problems presented in http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016,http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra, http://www.slideshare.net/JamesSmith245/solution-of-the-special-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra, and http://www.slideshare.net/JamesSmith245/a-very-brief-introduction-to-reflections-in-2d-geometric-algebra-and-their-use-in-solving-construction-problems. After reviewing, briefly, how reflections and rotations can be expressed and manipulated via GA, it solves the LLP limiting case of the Problem of Apollonius in two ways.
The document discusses key concepts about sets, including:
1) Intervals are subsets of real numbers that can be open, closed, or half-open/half-closed. Intervals are represented visually on a number line.
2) The power set of a set A contains all possible subsets of A. Its size is 2 to the power of the size of A.
3) The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements common to both sets.
4) Practical problems can use formulas involving the sizes of unions and intersections of finite sets.
1. Material implication is a rule of replacement in symbolic logic that states a disjunction or conditional statement is logically equivalent to its reverse conditional statement.
2. Material implication can be applied to either the whole line or part of a line by reversing any negations and changing conjunctions to disjunctions or vice versa.
3. Several examples are provided to demonstrate applying material implication to derive new lines, including to set up hypothetical syllogisms and modus ponens arguments.
Solution Strategies for Equations that Arise in Geometric (Clifford) AlgebraJames Smith
Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this document presents, explains, and discusses common solution strategies. Included are a list of formulas and a guide to nomenclature.
See also:
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016 ;
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial ;
http://www.slideshare.net/JamesSmith245/solution-of-the-special-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra ;
http://www.slideshare.net/JamesSmith245/a-very-brief-introduction-to-reflections-in-2d-geometric-algebra-and-their-use-in-solving-construction-problems ;
http://www.slideshare.net/JamesSmith245/solution-of-the-llp-limiting-case-of-the-problem-of-apollonius-via-geometric-algebra-using-reflections-and-rotations ;
http://www.slideshare.net/JamesSmith245/simplied-solutions-of-the-clp-and-ccp-limiting-cases-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/additional-solutions-of-the-limiting-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/an-additional-brief-solution-of-the-cpp-limiting-case-of-the-problem-of-apollonius-via-geometric-algebra-ga .
This document provides an outline and examples for proving theorems related to midpoints and intercepts in triangles. It includes:
1. Definitions of parallel lines, congruent triangles, and similar triangles.
2. Examples of proofs of the Triangle Midpoint Theorem - which states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long.
3. An example proof of the Triangle Intercept Theorem - which states that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.
This document contains a mathematics exam for Secondary School students in Perak, Malaysia. It covers topics like sets, Venn diagrams, linear inequalities, simultaneous linear equations, quadratic equations and expressions, solid geometry, and mathematical reasoning. There are 10 multiple choice questions for each topic area testing students' understanding of key concepts and ability to solve related problems. The document is in Malay and contains diagrams to illustrate the questions.
it is the first Homework.
it is about..
1-)The Foundations: Logic and Proofs
2-)Basic Structures: Sets, Functions, Sequences, Sums, and Matrices
3-)Number Theory and Cryptography
4-)Induction and Recursion
1. In Boolean algebra, a variable represents a logical quantity that can have a value of 1 or 0. Operations like addition and multiplication represent logical OR and AND operations.
2. Karnaugh maps are used to simplify Boolean expressions by grouping variables and eliminating variables that change between adjacent cells. This groups variables to find the minimum logic expression.
3. Hardware description languages like VHDL and Verilog allow digital designs to be described and implemented using code. VHDL uses entities to describe inputs and outputs, and architectures to describe logic, while Verilog uses modules.
This document introduces some basic concepts of set theory:
1. Sets are collections of well-defined objects that can be represented using capital letters. Elements of a set are denoted using symbols like ∈ and ∉.
2. Important sets in number systems include real numbers (IR), positive/negative reals (IR+/IR-), integers (Z), positive/negative integers (Z+/Z-), rational numbers (Q), and natural numbers (N).
3. Sets can be specified using a roster method that lists elements or a set-builder notation that describes elements. Operations on sets include union, intersection, complement, and symmetric difference.
This document provides an overview of matrix algebra concepts including:
- Matrix addition is defined as adding corresponding elements and is commutative and associative.
- Matrix multiplication is defined as taking the dot product of rows and columns. It is associative but not commutative.
- The transpose of a matrix is obtained by flipping rows and columns.
- Properties of matrix operations like addition, multiplication, and transposition are discussed.
This document provides an introduction to ordered binary decision diagrams (OBDDs). It discusses how OBDDs provide a canonical form for representing propositional logic formulas as decision trees, with sharing of identical subtrees. It describes how OBDDs can efficiently detect if a formula is a tautology or inconsistent. The document outlines the algorithm for converting a formula to an OBDD in canonical form, respecting a given variable ordering and sharing subtrees. It notes various optimizations that can be performed based on using hash tables to avoid constructing or processing the same subtrees multiple times.
1. Basic Engineering
Truth Tables and Boolean Algebra
F Hamer, R Horan & M Lavelle
The aim of this document is to provide a short,
self assessment programme for students who
wish to acquire a basic understanding of the
fundamentals of Boolean Algebra through the
use of truth tables.
Copyright c
2005 Email: chamer, rhoran, mlavelle@plymouth.ac.uk
Last Revision Date: May 18, 2005 Version 1.0
2. Table of Contents
1. Boolean Algebra (Introduction)
2. Conjunction (A ∧ B)
3. Disjunction (A ∨ B)
4. Rules of Boolean Algebra
5. Quiz on Boolean Algebra
Solutions to Exercises
Solutions to Quizzes
The full range of these packages and some instructions,
should they be required, can be obtained from our web
page Mathematics Support Materials.
3. Section 1: Boolean Algebra (Introduction) 3
1. Boolean Algebra (Introduction)
Boolean algebra is the algebra of propositions. Propositions will be
denoted by upper case Roman letters, such as A or B, etc. Every
proposition has two possible values: T when the proposition is true
and F when the proposition is false.
The negation of A is written as ¬A and
read as “not A”. If A is true then ¬A is
false. Conversely, if A is false then ¬A is
true. This relationship is displayed in the
adjacent truth table.
A ¬A
T F
F T
The second row of the table indicates that if A is true then ¬A is false.
The third row indicates that if A is false then ¬A is true. Truth tables
will be used throughout this package to verify that two propositions
are logically equivalent. Two propositions are said to be logically
equivalent if their truth tables have exactly the same values.
4. Section 1: Boolean Algebra (Introduction) 4
Example 1
Show that the propositions A and ¬(¬A) are logically equivalent.
Solution
From the definition of ¬ it follows that if
¬A is true then ¬(¬A) is false, whilst if
¬A is false then ¬(¬A) is true. This is
encapsulated in the adjacent truth table.
A ¬A ¬(¬A)
T F T
F T F
From the table it can be seen that when A takes the value true, the
proposition ¬(¬A) also takes the value true, and when A takes the
value false, the proposition ¬(¬A) also takes the value false. This
shows that A and ¬(¬A) are logically equivalent, since their logical
values are identical.
Logical equivalence may also be written as an equation which, in this
case, is
A = ¬(¬A) .
5. Section 2: Conjunction (A ∧ B) 5
2. Conjunction (A ∧ B)
If A and B are two propositions then the “conjunction” of A and B,
written as A ∧ B, and read as “A and B ”, is the proposition which is
true if and only if both of A and B are true. The truth table for
this is shown.
There are two possible values for
each of the propositions A, B, so
there are 2 × 2 = 4 possible assign-
ments of values. This is seen in the
truth table. Whenever truth tables
are used, it is essential that every
possible value is included.
A B A ∧ B
T T T
T F F
F T F
F F F
Truth table for A ∧ B
Example 2 Write out the truth table for the proposition (A∧B)∧C.
Solution The truth table will now contain 2 × 2 × 2 = 8 rows, cor-
responding to the number of different possible values of the three
propositions. It is shown on the next page.
6. Section 2: Conjunction (A ∧ B) 6
In the adjacent table, the
first three columns con-
tain all possible values for
A, B and C. The values
in the column for A ∧ B
depend only upon the first
two columns. The val-
ues of (A ∧ B)∧C depend
upon the values in the
third and fourth columns.
A B C A ∧ B (A ∧ B) ∧ C
T T T T T
T T F T F
T F T F F
T F F F F
F T T F F
F T F F F
F F T F F
F F F F F
Truth table for (A ∧ B) ∧ C
Exercise 1. (Click on the green letters for the solutions.)
(a) Write out the truth table for A ∧ (B ∧ C).
(b) Use example 2 and part (a) to prove that
(A ∧ B) ∧ C = A ∧ (B ∧ C) .
7. Section 3: Disjunction (A ∨ B) 7
3. Disjunction (A ∨ B)
If A and B are two propositions then the “disjunction” of A and B,
written as A ∨ B, and read as “A or B ”, is the proposition which is
true if either A or B, or both, are true. The truth table for this is
shown.
As before, there are two possi-
ble values for each of the propo-
sitions A and B, so the number
of possible assignments of values
is 2 × 2 = 4 . This is seen in the
truth table.
A B A ∨ B
T T T
T F T
F T T
F F F
Truth table for A ∨ B
Example 3 Write out the truth table for the proposition (A∨B)∨C.
Solution The truth table will contain 2×2×2 = 8 rows, correspond-
ing to the number of different possible values of the three propositions.
It is shown on the next page.
8. Section 3: Disjunction (A ∨ B) 8
In the adjacent table, the
first three columns con-
tain all possible values for
A, B and C. The values
in the column for A ∨ B
depend only upon the first
two columns. The val-
ues of (A ∨ B)∨C depend
upon the values in the
third and fourth columns.
A B C A ∨ B (A ∨ B) ∨ C
T T T T T
T T F T T
T F T T T
T F F T T
F T T T T
F T F T T
F F T F T
F F F F F
Truth table for (A ∨ B) ∨ C
Exercise 2. (Click on the green letters for the solutions.)
(a) Write out the truth table for A ∨ (B ∨ C).
(b) Use example 3 and part (a) to show that
(A ∨ B) ∨ C = A ∨ (B ∨ C) .
9. Section 3: Disjunction (A ∨ B) 9
Exercise 3. Write out the truth tables for the following propositions.
(Click on the green letters for the solutions.)
(a) ¬(A ∧ B), (b) (¬A) ∧ B, (c) (¬A) ∧ (¬B),(d) (¬A) ∨ (¬B),
Quiz Using the results of exercise 3, which of the following is true?
(a) (¬A) ∧ B = (¬A) ∨ (¬B), (b) ¬(A ∧ B) = (¬A) ∨ (¬B),
(c) ¬(A ∧ B) = (¬A) ∧ (¬B), (d) (¬A) ∧ B = (¬A) ∧ (¬B).
Exercise 4. Write out the truth tables for the following propositions.
(Click on the green letters for the solutions.)
(a) A ∧ (B ∨ C), (b) (A ∧ B) ∨ C, (c) (A ∧ B) ∨ (A ∧ C).
Quiz Using the results of exercise 4, which of the following is true?
(a) A ∧ (B ∨ C) = (A ∧ B) ∨ C, (b) A∧(B∨C) = (A∧B)∨(A∧C),
(c) (A∧B)∨C = (A∧B)∨(A∧C),(d) None of these.
The three different operations ¬ , ∧ , and ∨ , (“not”, “and” and “or”),
form the algebraic structure of Boolean algebra. What remains to be
determined is the set of rules governing their interactions.
10. Section 4: Rules of Boolean Algebra 10
4. Rules of Boolean Algebra
(1a) A ∧ B = B ∧ A
(1b) A ∨ B = B ∨ A
(2a) A ∧ (B ∧ C) = (A ∧ B) ∧ C
(2b) A ∨ (B ∨ C) = (A ∨ B) ∨ C
(3a) A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C)
(3b) A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
(4a) A ∧ A = A
(4b) A ∨ A = A
(5a) A ∧ (A ∨ B) = A
(5b) A ∨ (A ∧ B) = A
(6a) A ∧ ¬A = F
(6b) A ∨ ¬A = T
(7) ¬(¬A) = A
(8a) ¬(A ∧ B) = (¬A) ∨ (¬B)
(8a) ¬(A ∨ B) = (¬A) ∧ (¬B)
11. Section 4: Rules of Boolean Algebra 11
The following comments on these rules are useful:
All the rules can be verified by using truth tables.
Any rule labelled (a) can be obtained from the corresponding rule
labelled (b) by replacing ∧ with ∨ and ∨ with ∧ , and vice versa.
Rules (1a) and (1b) can easily be shown to be true.
Rules (2a) and (2b) were proved in exercises 1 and 2 respectively.
Rule (3a) was proved in exercise 3 and the quiz immediately following.
Rules (3a) and (3b) are called the distributive rules.
Rule (8a) was proved in exercise 4 and the quiz immediately following.
Rules (8a) and (8b) are called De Morgan’s laws.
Rule (2a) means that A ∧ B ∧ C have the same meaning given by
A ∧ (B ∧ C) or (A ∧ B) ∧ C. Similarly for A ∨ B ∨ C from (2b).
12. Section 4: Rules of Boolean Algebra 12
Exercise 5. Use truth tables to prove the following propositions.
(Click on the green letters for the solutions.)
(a) A∨(B∧C) = (A∨B)∧(A∨C), (b) A ∧ (A ∨ B) = A,
(c) ¬(A ∨ B) = ¬(A) ∧ ¬(B), (d) F ∨ X = X,
(e) [A ∧ ((¬A) ∨ B)] ∨ B = B.
Truth tables are useful for proving that two expressions are equivalent
but, often, the same result is easier to obtain using Boolean algebra.
Example 4 Simplify the expression [A ∧ (¬A ∨ B)] ∨ B.
Solution
Using (3a),
A ∧ (¬A ∨ B) = (A ∧ ¬A) ∨ (A ∧ B) = F ∨ (A ∧ B) , by (6a) ,
= A ∧ B , by ex 5(d) .
Thus [A ∧ (¬A ∨ B)] ∨ B = (A ∧ B) ∨ B = B , by (5b) ,
confirming the result of exercise 5(e), above.
13. Section 5: Quiz on Boolean Algebra 13
5. Quiz on Boolean Algebra
Begin Quiz In each of the following, choose the simplified version of
the given expression. (Use either truth tables, or Boolean algebra to
simplify the expression.)
1. (A ∧ ¬C) ∨ (A ∧ B ∧ C) ∨ (A ∧ C).
(a) T, (b) F, (c) A ∧ B, (d) A.
2. [A ∧ B] ∨ [A ∧ ¬B] ∨ [(¬A) ∧ B] ∨ [(¬A) ∧ (¬B)].
(a) A ∧ B, (b) A ∨ B, (c) T, (d) F.
3. (A ∧ B ∧ C) ∨ (¬A) ∨ (¬B) ∨ (¬C).
(a) T, (b) F, (c) A ∧ B, (d) A ∧ C.
End Quiz
14. Solutions to Exercises 14
Solutions to Exercises
Exercise 1(a)
The truth table for A ∧ (B ∧ C) is constructed as follows:
In the first three columns,
write all possible values
for the propositions A , B
and C, and use these to
calculate the values in the
fourth column. The final
column is found by taking
the “conjunction” of the
first and fourth columns.
A B C B ∧ C A ∧ (B ∧ C)
T T T T T
T T F F F
T F T F F
T F F F F
F T T T F
F T F F F
F F T F F
F F F F F
Click on the green square to return
15. Solutions to Exercises 15
Exercise 1(b)
The logical equivalence of the propositions (A∧B)∧C and A∧(B∧C)
can be seen by comparing their truth tables. The former is given in
example 2 and the latter in exercise 1(a). These truth tables are
identical, so
(A ∧ B) ∧ C = A ∧ (B ∧ C) .
Click on the green square to return
16. Solutions to Exercises 16
Exercise 2(a)
The truth table for A ∨ (B ∨ C) is shown below:
The first three columns
contain all possible values
for the propositions A, B
and C. The fourth col-
umn represents the “dis-
junction” (B ∨ C) and
the last column shows the
“disjunction” of elements
from the first and fourth
columns.
A B C (B ∨ C) A ∨ (B ∨ C)
T T T T T
T T F T T
T F T T T
T F F F T
F T T T T
F T F T T
F F T T T
F F F F F
Click on the green square to return
17. Solutions to Exercises 17
Exercise 2(b)
To prove the logical equivalence
(A ∨ B) ∨ C = A ∨ (B ∨ C)
compare the truth tables for both of these propositions. The former
is given in example 3 and the latter in exercise 2(a). These truth
tables are identical and therefore (A ∨ B) ∨ C = A ∨ (B ∨ C).
Click on the green square to return
18. Solutions to Exercises 18
Exercise 3(a)
The truth table for the proposition ¬(A ∧ B) is given below.
The adjacent truth table
contains 2×2 = 4 rows cor-
responding to all possible
values for the propositions
A and B.
A B A ∧ B ¬(A ∧ B)
T T T F
T F F T
F T F T
F F F T
Click on the green square to return
19. Solutions to Exercises 19
Exercise 3(b)
The truth table for the proposition (¬A) ∧ B is given below.
In the adjacent truth table
the third column contains
the “negation” of a given
proposition A and the last
column represents its “con-
junction” with the proposi-
tion B.
A B ¬A (¬A) ∧ B
T T F F
T F F F
F T T T
F F T F
Click on the green square to return
20. Solutions to Exercises 20
Exercise 3(c)
The truth table for the proposition (¬A) ∧ (¬B) is given below.
In the truth table the third
and fourth columns con-
tain the “negation” of the
propositions A and B re-
spectively, the last column
is the “conjunction” of ¬A
and ¬B.
A B ¬A ¬B (¬A) ∧ (¬B)
T T F F F
T F F T F
F T T F F
F F T T T
Click on the green square to return
21. Solutions to Exercises 21
Exercise 3(d)
The truth table for the proposition (¬A) ∨ (¬B) is below.
In this truth table the
third and fourth columns
contain the “negation” of
propositions A and B re-
spectively, while the last
column is the “disjunction”
of ¬A and ¬B.
A B ¬A ¬B (¬A) ∨ (¬B)
T T F F F
T F F T T
F T T F T
F F T T T
Click on the green square to return
22. Solutions to Exercises 22
Exercise 4(a)
The truth table for the proposition A ∧ (B ∨ C) is below.
In the adjacent table, the
first three columns contain
all 8 possible values for
propositions A, B and C.
The fourth column shows
the “disjunction” of B and
C, while the last one is
A ∧ (B ∨ C).
A B C B ∨ C A ∧ (B ∨ C)
T T T T T
T T F T T
T F T T T
T F F F F
F T T T F
F T F T F
F F T T F
F F F F F
Click on the green square to return
23. Solutions to Exercises 23
Exercise 4(b)
The truth table for the proposition (A ∧ B) ∨ C is below.
The first three columns
contain all possible values
for propositions A, B and
C. The fourth column
shows the “conjunction”
of A and B, while the last
one is (A ∧ B) ∨ C.
A B C A ∧ B (A ∧ B) ∨ C
T T T T T
T T F T T
T F T F T
T F F F F
F T T F T
F T F F F
F F T F T
F F F F F
Click on the green square to return
24. Solutions to Exercises 24
Exercise 4(c)
The truth table for the proposition (A ∧ B) ∨ (A ∧ C) is below.
The fourth and
fifth columns
show the “con-
junction” of A
and B, and A
and C, respec-
tively. The final
column is the
“disjunction” of
these.
A B C A ∧ B A ∧ C (A ∧ B) ∨ (A ∧ B)
T T T T T T
T T F T F T
T F T F T T
T F F F F F
F T T F F F
F T F F F F
F F T F F F
F F F F F F
Click on the green square to return
25. Solutions to Exercises 25
Exercise 5(a) The truth tables for A∨(B ∧C) and (A∨B)∧(A∨C)
are given below.
A B C A ∨ (B ∧ C)
T T T T
T T F T
T F T T
T F F T
F T T T
F T F F
F F T F
F F F F
A B C (A ∨ B) ∧ (A ∨ C)
T T T T
T T F T
T F T T
T F F T
F T T T
F T F F
F F T F
F F F F
Comparing them shows that
A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C) .
Click on the green square to return
26. Solutions to Exercises 26
Exercise 5(b)
The truth table for (A ∨ B) is given below.
A B A ∨ B A ∧ (A ∨ B)
T T T T
T F T T
F T T F
F F F F
Comparing the first and the last columns above shows that
A ∧ (A ∨ B) = A .
Click on the green square to return
27. Solutions to Exercises 27
Exercise 5(c)
The truth table for ¬(A ∨ B) and ¬(A) ∧ ¬(B) are given below.
A B ¬(A ∨ B)
T T F
T F F
F T F
F F T
A B (¬A) ∧ (¬B)
T T F
T F F
F T F
F F T
The truth tables are identical so these propositions are logically equiv-
alent, i.e.
¬(A ∨ B) = ¬(A) ∧ ¬(B) .
Click on the green square to return
28. Solutions to Exercises 28
Exercise 5(d)
For an arbitrary proposition X, the “disjunction” of X and “false”
(i.e. the proposition F) has the following truth table.
F X F ∨ X
F T T
F F F
Therefore the relation
F ∨ X = X
is correct.
Click on the green square to return
29. Solutions to Exercises 29
Exercise 5(e)
For any two propositions A and B, construct the following truth table:
A B (¬A) ∨ B A ∧ ((¬A) ∨ B) [A ∧ ((¬A) ∨ B)] ∨ B
T T T T T
T F F F F
F T T F T
F F T F F
Comparing the last column with the second one we find that
[A ∧ ((¬A) ∨ B)] ∨ B = B .
Click on the green square to return
30. Solutions to Quizzes 30
Solutions to Quizzes
Solution to Quiz:
The truth tables from exercise 3(a) and exercise 3(d) are, respectively,
A B ¬(A ∧ B)
T T F
T F T
F T T
F F T
A B (¬A) ∨ (¬B)
T T F
T F T
F T T
F F T
Since these are identical, it follows that
¬(A ∧ B) = (¬A) ∨ (¬B) .
End Quiz
31. Solutions to Quizzes 31
Solution to Quiz:
The truth tables below are from exercise 4(a) and exercise 4(c), re-
spectively.
A B C A ∧ (B ∨ C)
T T T T
T T F T
T F T T
T F F F
F T T F
F T F F
F F T F
F F F F
A B C (A ∧ B) ∨ (A ∧ B)
T T T T
T T F T
T F T T
T F F F
F T T F
F T F F
F F T F
F F F F
From these, it can be seen that
A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C) .
End Quiz