Sentences And Solution Sets (1959)

the idea of set is one with which we are all acquainted in ordinary life we all recognize the properties common to a specific group of object the individual members that make up this set of pieces of chalk or this set of pencils all have something in common the first set that all the members are chalk the second that all the members of it are pencil examples can be found everywhere in the classroom a teacher could use a set of chalk or pencil or books or the desks the children sit at to get this concept across to his pupil the important thing is that we are able to tell of each object whether or not it belongs to the set we're talking about this same concept occurs in all branches of mathematics there are sets of numbers and sets of points as well as sets of books and pencils for example there is the set of even two-digit numbers or the set of all odd two-digit numbers or if we wanted to we could limit a set at will defining it for instance really as the three numbers one two and three until now this notion of set has not largely entered the high school classroom and it's time that it did because it offers three important advantages in our teaching it brings clarification putting our ideas into sharper focus it offers simplification the very language of sets helps us to eliminate vague terms and trim off excess verbiage finally it gives unification since the concept of set pervades all areas mathematics although formal set theory certainly should not be part of a high school course the concepts and vocabulary of sets presented quite informally in the classroom can arouse interest in students and promote fuller understanding for example let's visit a class whose teacher is introducing the idea of variable from this point of view we've spent some time discussing what a set is and I'm sure you all understand what we mean by it let's take an example our set which will designate by the letter S is all the states in the United States I'm going to make some statements about its members or elements and I would like you to tell me if these statements are true or not the state of Virginia is one of the original 13 is that true Joan why yes if I take out the word Virginia and replace it with a blank is that true it isn't true or anything now in other words it's neither true nor false until I replaced that blank with one of the elements of the set we're talking about this is what we call an open sentence because well it's wide open isn't it if I put the word New York in there is that true Barbara it's true to reinforce the concept of an open sentence our teacher uses various replacements for the blank space some true some false then he can introduce the idea of using a letter as a placeholder there I have a sentence and it has one variable that we're calling Z regardless of whether the sentence is true or false what can we replace Z by the name of any one of the United States that's good any one of the elements of our set can be the replacement can it yes well it will be true only thirteen times that's true but the variable here Z would still be a placeholder or one name selected from the collection of possible replacements one element of our set which in this case is all of the states in the Union yes Richard I understand this but what does it have to do with mathematics in mathematics instead of sets of states the sets are often made up of numbers or numbers no only those numbers of the set that we have decided to use as replacements we call this the universal set and we use a capital u to designate it in my universe or replacement set is the numbers 1 2 3 4 5 6 & 7 it represents everything every member that we can use could I talk about an 8 no it's outside the universe Janus has the idea all right now here's an open sentence for you with one variable X plus 3 equals 7 suppose we try all the possible replacements for our variable X 1 plus 3 equals 7 but that's not true mister me you any correct and so we'll do this let's keep all the false statements in one column shall we mr. me Winnie has his class substitute values from the universal set 4 plus 3 equals 7 now that's true mr. mere any a new column is made up for the one true statement 4 plus 3 equals 7 and then the class continues until it has substituted all possible values from the universal set well have we run out of possibilities for our variable sure Martha what numbers from our set did we use that gave us false answers 1 2 3 5 6 & 7 now that's a set of numbers they all have something in common what is it they all gave you false answer right and all the members of this set were also members of you and so let's call it a subset of you I'm saving subset 1 for the true answers now Barbara is there a set of numbers that gave us true answers only one the number 4 can that be if they're all by itself indeed it can and so we have another subset of you that consists of the one number four now we have two subsets of the universal set you on what basis did we divide the universal set into two subsets why did we put some numbers here under false and one number here under true well it's all whether or not X plus 3 equals 7 that's right Janice this statement here really selected the two subsets didn't it yes and that's why we call it the set selector now class which of these two subsets gives us the solution to our equation s1 the truth de good so we'll just call s1 the solution set because it's a set of replacements that makes the sentence true now let's explore some of these ideas we've seen discussed first we'll talk about solution sense the replacement or Universal sets used in high school algebra are usually all or part of these number systems integers rational numbers real numbers and complex numbers it's most important that we keep in mind which of these systems is the universal set for the particular problem that we have under consideration it makes a great difference for example consider the expression x squared minus 2 if we are working in the field of rational numbers we would say that this expression cannot be factored it has no factors with rational coefficients however if we are working in the field of real numbers we could write these factors let's look at another example this time we shall take an inequality as our sentence in one variable the possible replacements for the variable X will be determined by the universal set or the replacement set that we specify if u is the set of rational numbers the replacements will include all possible whole numbers and fractions but if u is the set of all positive integers the possible replacements are limited to these integers here the sentence in one variable X is greater than or equal to 5 is the set selector the set of positive integers for which the sentence is true is 5 6 7 8 and so on indefinitely this is the solution set we can picture this graph on the number line will indicate the elements of the replacement or universal set by small circles the solution set is indicated by the dotted circle a convenient notation is this the vertical line stands for such that and the whole statement is red the set of all positive integers such that X is greater than or equal to 5 is the set consisting of 5 6 7 8 and so on the left side of the equation is called the set builder one of the changes in content being suggested for high school algebra by those interested in curricular reform is the inclusion of work on inequalities the notion of solution set is particularly helpful in teaching this work because the solutions of inequalities are often as in our last illustration sets of many members whereas equations at the 9th grade level generally offer solution sets consisting of only one or two numbers another reason for the inclusion of inequalities is that they offer a large variety of possible open sentences for example we can discuss the sentence X is greater than 5 we can make a new sentence by merely adding an equal sign we can change to X is not greater than 5 or we can change to X is not greater than or equal to 5 this offers a variety of problems arising from the same situation in which the notion of the solution set can be developed even in dealing with equations how one can express the facts commonly taught about solutions very simply in set language for example a quadratic equation has precisely two real or imaginary roots which may be equal this fact commonly taught in 11th grade algebra may be expressed by saying that the solution set has two members as indicated earlier the language of sets office simplification as well as clarification so much for these notions the language of sets offers unification equation and that of an inequality the notion of variable itself a good example of how the language of sets clarifies ideas that might otherwise be fuzzy in mr. Mahoney's class students learned that a variable is a placeholder a symbol in a sentence that holds a place for or may be replaced by the elements of a given set this is sharp and precise but try saying it without using the word set either the statement is incomplete or it is vague and inaccurate we have to use descriptions that are far from precise such as a general number a literal number or a changing number these same ideas are applicable to sentences that do not contain numerical variable for example consider this sentence at various times from 1787 to the present the replacement set was differently constituted and at different times different states or the largest from 1845 until 1958 the solution was Texas in 1958 Congress changed the replacement set thereafter the solution became Alaska and at different times between 1787 and 1845 this sentence at different solutions the solution depended on what the replacement set was at the time that is what states were in the Union at the time the solution was soft now let's go back to the idea of set-builder this has pointed out is really a notation but it is a particularly fruitful notation it enables us easily to talk about all the X's or rather the set of all X's that satisfy some specified condition and determining all the X's that satisfy some condition is precisely what we do when we solve an algebraic problem here are a few examples the set of all X's such that X square minus 5x plus 6 equals 0 is the set consisting of the numbers 3 & 2 the set of all X's such that 3 X square minus 20x minus 7 equals 0 is the set 7 minus one-third 4 3 X square minus 20x minus 7 equals 3x plus 1 times X minus 7 the set of all X's such that X square is less than 4 is all the numbers between minus 2 and plus 2 not inclusive the set of all X's such that X is divisible by 3 and also less than 20 is the set consisting of the six numbers 3 6 9 12 15 18 all of them are suitable for classroom use once the novelty of the language and the notation has worn off it will indeed be found that these techniques do offer clarification simplification unification