Hermine Smikle.
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 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:
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(a) New Haven is a city in Connecticut.
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(b) The month of June has thirty days.
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(c) The moon is made of red cheese.
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(d) Tomorrow is Saturday.
The following are not statements:
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(a) Come to our party!
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(b) Is your homework done?
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(c) Close the door when you leave.
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(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
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(a) Three is a counting number.
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(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.
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Example
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(a) Ann is early for class and she has her note books.
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(b) Three is a counting number and is also a odd number.
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Types of Compound statements and their connectives
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1. A negation: formed when we negate a simple statement by “not”.
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example : Simple statement: Today is Thursday
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Compound statement: negation: today is not Thursday
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The sentence “today is not Thursday” is a compound statement called a negation.
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2. When we connect two simple statements using and, the result is a compound statement called a conjunction.
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3. If the simple statements are joined by or the resulting compound statement is called a disjunction.
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4. The If . . . then connector is used in compound statements called conditionals.
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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.
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Examples.
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P = Today is Saturday
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Q = I passed my test
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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)
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Connectors
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Symbols
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(a) not
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~
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(b) and
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^
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(c) or
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(figure available in print form)
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(d) if . . . then
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Ð>
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(e) if and only if
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Ð>
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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.
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Example
1: Let P = this is a hard course.
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~P= this is not a hard course.
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Truth Table
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p
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~p
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where
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T = True and
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T
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F
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F = false
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F
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T
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In the first column, there are two possibilities of P; P is either True or False. Each line in the table represents a case that must be considered. In this case, there are only two cases. The truth table tells us the truth value of p in every case.
Truth Tables with the Connective
^
The Connective ^ may be placed between any two statements P and Q to form the compound statement p^q
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Let
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P = Today is Monday
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Q = I have a Math class.
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Truth Table
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P
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Q
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P^Q
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T
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T
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T
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T
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F
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F
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F
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T
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F
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F
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F
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F
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In the compound statements, the individual statements are called components. In a compound statement with two components such as p ^ q there are four possibilities. These are called logical possibilities.
The possibilities are:
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1) p is true and q is true
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2) p is true and q is false
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3) p is false and q is true
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4) p is false and q is false.
The four possibilities are covered in the four rows of the truth table. The last column gives values of p ^ q; This is only true when both p and q are true.
Using the examples given, truth tables of a more complicated nature can be built.
Let us consider the situation p v q
Example
2:
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P
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Q
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P v Q
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P = Today is Tuesday
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T
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T
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T
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Q = I have a Math class
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F
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T
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T
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P v Q = Today is Tuesday or I have a math class
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F
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F
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T
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F
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F
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F
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Find the Truth Table of [ ( p) ^ ( q) ]
Example
3:
Truth Table
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p
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q
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~p
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~q
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(~p) ^ ~ q
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~ [(~p) ^ (~q)]
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T
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T
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F
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F
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F
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T
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T
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F
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F
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T
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F
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T
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F
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T
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T
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F
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F
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T
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F
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F
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T
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T
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T
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F
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The given [ (~p) ^ (~q) ] uses parentheses and brackets to indicate the order in which the connectives apply.
Expressions can be simplified by removing some of the parentheses, thus (p) ^ ( q) can be written as ~ p ^~ q.
It can be noticed from the Truth Table in examples 2 and 3 that the last columns are the same. Thus we say that these statements are logically equivalent and can be written P = Q and P v Q = (~p ^ ~q).
The conditional and the Biconditionals Statements.
If the connectors Ð> is used between any two statements P and Q to form a compound statement P Ð> Q (reads if P then Q), the statement is called a conditional statement.
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Let
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P = You passed English
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Q = You will graduate
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Truth Table
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P
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Q
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PÐ>Q
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T
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T
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T
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T
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F
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F
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F
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T
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T
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F
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F
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T
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The statement P Ð> Q reads if you pass English then you will graduate. This statement is false only when you pass English (true) but you will not graduate. Therefore the final column will be true in every position but the second.
The connective Ð> is called the biconditional and may be placed between any two statements to form a compound statement P Ð> Q (reads P if and only if Q).
The Truth Table For P Ð> Q
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P
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Q
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P Ð> Q
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PÐ> Q
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QÐ> P ( P (Ð> Q)
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^ ( Q Ð> P)
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T
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T
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T
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T
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T
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T
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T
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F
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F
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F
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T
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F
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F
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T
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F
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T
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F
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F
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F
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F
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T
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T
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T
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T
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From the truth table it can be noticed that P Ð> q = ( P Ð> Q) ^ (QÐ>P).
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Sample Problems for Students:
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1. In these problems English sentences are given. In each case determine whether the sentence is a statement or not.
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(a) On March 8, 1922 snow fell in Atlanta.
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(b) Mary has big feet.
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(c) How much did you pay for that car.
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(d) Keep off the grass.
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(e) Five is a prime Number.
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2. If you accept the sentences in column 1 what can you say about the statements in column 2
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Column 1
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Column 2
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(a) The order of 1, 2, 3, 4, 5, 6 on
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The relationship between 5 and 2
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number line
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(b) That a head and a tail are
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The number of heads most likely
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equally likely on the toss
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to be obtained in 600 throws
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of a coin
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(c) That a parallelogram can be
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The figure ABCD is a parallelogram
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formed two congruent triangles
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(d) If all policemen are over
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Mr brown is a policeman
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six feet tall.
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(e) A person must be 16 years to
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Junior is driving a car
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drive a car
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3. Make a truth table for the given statements.
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(a) ~(~P)
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(b) ~PÐ> ~ Q
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(c) ~ PÐ> Q
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(d) P ^ QÐ>P
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(e) ~ pÐ>Q
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(f) ( PÐ>Q) v PÐ>Q
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(g) P ^ Q P v Q
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(g) (PVQ) ^ R
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(h) (P ^ Q) v (P ^ Q).
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4. Construct truth table for
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(a) ~ (p ^ Q)
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(b) PV~Q
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(c) ~ ( Pv~Q)
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