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Boolean Algebra and Its Application to Problem Solving and Logic Circuits

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Boolean Algebra and Its Application to Problem Solving and Logic Circuits Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 1989 Volume VII: Electricity Boolean Algebra and its Application to Problem Solving and Logic Circuits Curriculum Unit 89.07.07 by Hermine Smikle. Purpose The curriculum unit is designed to introduce a unit of simple logic and have students exposed to the area of Boolean algebra and how it can be used as a tool for problem solving. These mathematical ideas have been left out of the curriculum of many high school students. There is a need for mathematics to become more relevant to today’s society. Boolean Algebra with its application to the development of logic circuits will provide some hands on application and to bring to students the importance of technology and the effects it has had on our society. Rationale of the unit The guiding principles of this unit is that any area of mathematics can be presented to students at the high school level and that mathematics should be a) relevant to the existing and growing needs of the society b) related to the ability of the student c) should serve as a motivator to careers that will interest the student d) provide the motivation for further inquiry. The basic principle of the unit shows how an area of mathematics and some fundamental concepts are applicable to real life and modern Technology. Students should be encouraged to seek out areas in which this technology apply and make some comparisons of the technology now and those that existed in years past. Objectives of the unit Curriculum Unit 89.07.07 1 of 26 a) To help students acquire a range of mathematical skills b) To make mathematics relevant to the experiences of the students, there-fore recognizing mathematical principles in his environment. c) To apply mathematical knowledge to the solution of problems. d) To present a unit in the area of Boolean Algebra that is important to the understanding of circuits and how they work. Audience : This unit is designed for classes at the high school level. It can be a part of a unit in geometry, or could be used by a group of students for independent study. The Specific Objectives of the Unit 1 Introduction to logic: ____ The students will be able to: ____ a) Distinguish valid statements from those that are not valid. ____ b) To define a statement; To use reasoning; make deductions and implications. ____ c) To use simple connectives. ____ d) To make truth tables. 2. Introduction to Binary Arithmetic: ____ a) Define the numerals in the Binary system. ____ b) To find the values of a numeral written in the binary system in base ten. ____ c) To perform the basic operations in the binary system. 3. Introduction to Boolean Algebra. ____ a) Simple operation with boolean algebra ____ b) Make truth tables ____ c) Application of Boolean Algebra to ____ ____ (1) electrical switches ; (using ON and OFF) ____ ____ (2) write truth for adders and half adders ____ ____ (3) designing simple circuits. Curriculum Unit 89.07.07 2 of 26 Introduction There has been much talk in the media about the expansion of the Japanese technology in the market place. Evidence can be seen in almost all aspect of our daily life. In the last ten years there have been the emergence of new appliances and gadgets that most people find too complicated to understand examples are the video recorder; CD players; cars that are more efficient and telephones that are more intelligent these are only a few. It will be the teachers responsibility to awaken the minds of these students to the great demand that will be placed on them to understand and be aware of these changing technologies and develop in the students a pest for knowledge, and the ability to seek new and different ways to solve problems. The work world of the next ten years will be demanding workers that are equipped with different basic skills; workers with the ability to think and can understand the operations of these machines developed by the new technology. The unit attempt to show students how the application of Boolean algebra, and the binary system has spearheaded work in these new technologies. After the unit it is hoped that students will reconsider the options available to them and make more careful and informed decisions as to their career choices. The unit will begin by discussing the implications of Reasoning and deduction in the formal setting, with extensive work in the binary system and then a simple introduction to boolean algebra. It is hoped that this unit will find a place with those teachers that are theorists, and those that enjoy working with the hands on experiences that are meaningful for students. Limitations of the unit The concepts of Boolean algebra are found in algebra texts designed for higher education students; therefore the language and symbolism used are very technical. In an attempt to make it appropriate most of the proofs will be omitted and only those concepts necessary for understanding will be used. Students should be encouraged to draw diagrams and make tables where necessary. Because of the limitation of space for the unit there will be the need for the users to research additional problems from the reference given. Reasoning and Deduction : Introduction to Logic The main ingredient in the study of logic is the principles and method used to distinguish between arguments that are valid and those that are not. Logic deals with reasoning and the ability to deduce or come to some reasonable conclusions. In everyday life we guess what is going to happen on the basis of past experiences; “It looks like its going to rain” we say meaning that it may rain today. If we wait around long enough then it may rain. This is an example of inductive reasoning. In mathematics we can discover whether or not a guess is correct by checking if our conclusions can be deduced from results already known. This is called deductive reasoning. The starting point of logic is a statement. A statement in the technical sense is declarative and is either true or false, but cannot be both simultaneously. In logic it is irrelevant whether a statement is true or false, the important thing is that it should be definitely Curriculum Unit 89.07.07 3 of 26 one or the other. Logic statements must be either true or false. A Statement: is a declarative sentence which is either true or false. Examples of declarative statements: (a) New Haven is a city in Connecticut. (b) The month of June has thirty days. (c) The moon is made of red cheese. (d) Tomorrow is Saturday. The following are not statements: (a) Come to our party! (b) Is your homework done? (c) Close the door when you leave. (d) Good by dear. Those are not good statements because they cannot be considered true or false. The basic type of sentence in logic is called a simple statement. A simple statement is one that has only one thought with no connecting word. Examples of simple statements (a) Three is a counting number. (b) Ann is early for class If we take a simple statement and join them with a connecting word such as and, or, if . then, not, if and only if, we form a new sentence called a complex or compound statement. Compound Statements: are formed from the combination of two or more simple statements. Curriculum Unit 89.07.07 4 of 26 (a) Ann is early for class and she has her Example note books. (b) Three is a counting number and is also a odd number. Types of Compound statements and their connectives 1. A negation: formed when we negate a simple statement by “not”. ____ example : Simple statement: Today is Thursday ____ Compound statement: negation: today is not Thursday ____ The sentence “today is not Thursday” is a compound statement called a negation. 2. When we connect two simple statements using and, the result is a compound statement called a conjunction. 3. If the simple statements are joined by or the resulting compound statement is called a disjunction. 4. The If . then connector is used in compound statements called conditionals. 5. The if and only if connector is used to form compound statements called biconditionals. We are familiar with using letters as replacements in algebra; in logic we can also use letters to replace statements. The common letters used to replace statements are P,Q, R: but any letters can be used. Examples. P = Today is Saturday Q = I passed my test but P and Q would read Today is Saturday and I passed my test. It is also common practice to use symbols for the connective words (or the connectors) Connectors Symbols (a) not ~ (b) and ^ (c) or (figure available in print form) (d) if . then Ð> Curriculum Unit 89.07.07 5 of 26 (e) if and only if Ð> TRUTH TABLES: Since a statement in logic is either true or false, we should be able to determine the truth or falsity of a given statement. [Logic is very precise. There should be no worry about ambiguity] Let P be a statement; then ~ P means ‘not P’ or the negation of P. The negation of P is true whenever the statement P is false and false if P is true. These situations are confusing to write, therefore we can record these statements in a truth table. Example 1: Let P = this is a hard course. ____ ____ ~P= this is not a hard course.
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