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The Game of Algebra or The Other Side of Arithmetic The Game of Algebra or The Other Side of Arithmetic © 2007 Herbert I. Gross by Herbert I. Gross & Richard A. Medeiros next Lesson 17 Part 1

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© 2007 Herbert I. Gross next Well-defined Sets Introduction to… Sets, Functions, and Graphs Introduction to… Sets, Functions, and Graphs

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Prologue © 2007 Herbert I. Gross next In the fable about the blind men and the elephant, each man touches the elephant; and based on which part he touches, each man gives a different description of what an elephant looks like. Each man was partly correct but not entirely correct.

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© 2007 Herbert I. Gross next A similar thing happens when we try to define mathematics. Namely, the subject is defined in different ways depending on who is using it and why. In our approach to this course we elected to use the concept of paraphrasing to introduce algebra. However, algebra is much more than just the art of paraphrasing statements the calculator can't “read” into equivalent statements that it can “read”. Among other things, it is the study of logical structure; and it is not limited to just the structure of number systems.

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© 2007 Herbert I. Gross next More specifically, the “game of mathematics” theme can be applied to structures whereby there are rules that tell us how to combine pairs of members in the group to obtain other members in the group. For example, Boolean Algebra (named after the British mathematician and logician, George Boole, who was born in 1815 and died in 1864) is a form of algebra concerned with logical structures where there are only two possible outcomes.

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© 2007 Herbert I. Gross -- In dealing with statements, a statement is either true or false; one or the other but not both. -- In dealing with electrical circuits, a switch is either open or closed (in other words, either on or off); one or the other but not both. -- In dealing with sets, something is either a member of the set or it isn't; one or the other but not both. For example next

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© 2007 Herbert I. Gross next In this context, Boolean algebra is fundamental to circuit design and to the design, function, and operation of modern computers. Our goal in this lesson is to develop a vocabulary that applies to any logical structure; not just to the arithmetic of numbers. In particular, part 1 deals with sets; part 2 with functions; and part 3 with graphs.

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© 2007 Herbert I. Gross next A common misconception that beginning students have is that they think of numbers as being devoid of structure. For example, when asked to define a rational number, a student might say that it's a symbol that contains a “top” part called the numerator and a “bottom” part called the denominator. However in fact, the common fraction symbol does not represent a rational number unless it also obeys the rules that rational numbers have to obey.

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© 2007 Herbert I. Gross next The building block of any structure is the set of objects for which the structure is going to be defined. To this end, we begin by defining sets from an objective rather than subjective point of view. That is, most of us know that a set is a collection of objects; such as a set of dishes, or a set of books.

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© 2007 Herbert I. Gross next However, the mathematical definition of a set is subtly different. Namely… A set is any collection for which there is a well-defined test for membership. In other words: the test has to be objective (rather than subjective). In such a case, the set is referred to as a well-defined set. However for brevity, we often just say “set” rather than “well-defined set”. The Mathematical Definition of a Set D

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© 2007 Herbert I. Gross next The set of beautiful paintings would not be a well-defined set because different people might have different interpretations of what it meant for a painting to be beautiful. In other words, the test for membership in this set is subjective (that is, it depends on the opinion of the observer). Several examples

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© 2007 Herbert I. Gross next The set of tall people would not be a well-defined set because different people might have different ideas concerning “how tall is tall”. For example, a person who might not look tall in the eyes of a person who was 6 feet tall might look very tall to a young child.

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© 2007 Herbert I. Gross On the other hand the set of all people who were at least 6 feet tall is well-defined because we can measure the height of each person and decide whether they meet the objective test for membership. next 6 f e e t

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© 2007 Herbert I. Gross next The set of all living United States Citizens whose birthday is celebrated on May 9 would be a well-defined set, since we could objectively check to see (1)whether the person was alive, (2) whether the person was a United States citizen, and (3) whether the person was born on May 9th. Any person who possessed these three properties would belong to the set and all other persons wouldn't.

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© 2007 Herbert I. Gross next The set of all even whole numbers is well-defined. That is, given any whole number, we can divide it by 2. If there is no remainder, the number belongs to the set; but if there is a remainder, it doesn't belong to the set

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© 2007 Herbert I. Gross next With respect to the set of living United States citizens who were born on May 9th, notice that the test for membership is objective, but we might have a difficult time finding all its members. So in the so called “new math” we invented vocabulary that we hoped would be more suggestive to students. In particular, the “new math” introduced two ways to describe a set.

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© 2007 Herbert I. Gross next It is possible that a set has infinitely many elements (such as the set of whole numbers). Other times the set might have a finite number of elements, but we don't know the specific names of them. For example, let's look at the set of all living Americans who were born on May 9, The Set Builder Notation for Describing a Set

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© 2007 Herbert I. Gross next In this case there is a definite, objective test for membership, but off hand we don't know the names of the members. In such a case, we enclose the set in braces: { }, and we pick a symbol to represent the elements of the set, say, x. The Set Builder Notation

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© 2007 Herbert I. Gross next We then write “x” inside the braces, followed by a colon {x: } and after the colon we write the test for membership. Thus, in this case we might write… {x: x is a living American who was born on May 9, 1929} We read the above as “the set of all x such that x is a living American who was born on May 9, 1929”.

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© 2007 Herbert I. Gross next In a similar way, suppose we wanted to use the set builder notation to represent the set of the New England states. We might denote the set by N and then write… N = {x:x is a New England state}

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© 2007 Herbert I. Gross next To help students focus on what it means to solve an equation such as 2x – 9 = 7, the “new math” introduced the idea of the solution set of an equation. For example, the solution set for the equation 2x – 9 = 7 was defined to mean the set of all numbers x for which 2x – 9 = 7; and the set builder notation was used to write the solution set in the form {x: 2x – 9 = 7}. An Application of the Set Builder Notation to Our Course

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© 2007 Herbert I. Gross next The beauty of using the set builder notation is that it tells us immediately what is meant by a solution. For example... If we call the solution set S, students could replace x by any number and check whether or not that number obeys the test for membership. Thus, if S = {x: 2x – 9 = 7} and we replace x by, say, 10; we would see that 2(10) – 9 ≠ 7 and hence, 10 is not a member of the solution set of the equation. However, 2(8) – 9 = 7 is a true statement, so 8 is a member of S.

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© 2007 Herbert I. Gross next It is rather lengthy to write the statement “8 is a member of the set S”. So mathematicians use the symbol є as an abbreviation for “is a member of” or “belongs to” and write “8 є S” rather than write “8 is a member of S ”. Some New Notation Then, in a way that is analogous to writing ≠ as an abbreviation for “is not equal to”, we use the abbreviation є to abbreviate “is not a member of”. Thus one might write 10 є S rather than “10 is not a member of S”. next

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© 2007 Herbert I. Gross next The term “solution set" replaced the “old” (i.e., before the 1950's) term “roots of the equation”. It also tightened the meaning of what it meant to solve an equation. The important thing, however, wasn’t whether we called them the roots or the solution set of the equation; but rather that we were able to paraphrase the set builder notation in a way that listed the members of the solution set explicitly; so this led to another way to describe sets.

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© 2007 Herbert I. Gross next Knowing the test for membership defines the members of the set implicitly but not explicitly. That is, we know the test for membership: but the test doesn't specifically tell us who these members are. 2. The Roster (or Listing) Method for Describing a Set.

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© 2007 Herbert I. Gross next The method for listing the members of a set is known as the roster method. In this method, we simply write the members of the set and enclose them in braces { }. This method is handy to use when there are relatively few members in the set, and we know what they all are. For example, the collection of New England states is a mathematical (well-defined) set because there is an objective test for membership.

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© 2007 Herbert I. Gross next The roster method for representing this set of New England states would be {Massachusetts, Rhode Island, Vermont, Connecticut, Maine, New Hampshire}. N = {Massachusetts, Rhode Island, Vermont, Connecticut, Maine, New Hampshire} Just as we did in the set builder notation, we often use a symbol, such as a letter (or letters) of the alphabet to denote a set, such as N (to stand for New England) and write…

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© 2007 Herbert I. Gross next We used N because it suggested the members of the set. However, just as with how we name variables in algebra, it makes no difference what symbol we use to denote the set. Thus, for example, we might have written… A = {Massachusetts, Rhode Island, Vermont, Connecticut, Maine, New Hampshire}

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© 2007 Herbert I. Gross next The set is determined by its members, not by the order in which its members are listed. Thus, for example, with N as defined above, we could also have written… N = {Vermont, Maine, Massachusetts, Rhode Island, Connecticut, New Hampshire}, etc. N = {New Hampshire, Rhode Island, Connecticut, Maine, Vermont, Massachusetts} or…

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© 2007 Herbert I. Gross next We have already discussed how algebra allows us to paraphrase expressions into equivalent forms that are easier for us to understand. In terms of the set builder notation and the roster notation, we may think of algebra as being the subject that allows us to “translate" the solution set from the set builder notation to the roster method. Application of the Roster Notation to Solving Algebraic Equations

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© 2007 Herbert I. Gross next For example, suppose we let S = {x: 2x – 9 = 7}. Since 2x – 9 = 7 implies that 2x = 16 and hence that x = 8, we may write the following string of equalities… S = {x: 2x – 9 = 7} = {x: 2x = 16} = {x: x = 8} so, S = {8} The above string of equalities shows how solving an algebraic equation is equivalent to transforming the implicit set builder form of the solution set to the explicit roster form of the solution set. next

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© 2007 Herbert I. Gross next When we write, for example, {x: 2x – 9 = 7} = {x: 2x = 16} we are not talking about the equality of two numbers, but rather about the equality of two sets. Notes on Equality of Two Sets Two sets, A and B, are called equal if they contain the same members. In other words A = B means that: every member of A is also a member of B, and every member of B is also a member of A. Thus, for example, {1, 2, 3} = {2, 3, 1}. next

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© 2007 Herbert I. Gross next One might wonder why we would give the same set two different names. In other words if A = B, why not just use A or B rather than both? One important reason is that two different tests for membership might both be describing the same set, but we might not know this at first glance. For example, suppose we let S = {x: 2x – 9 = 7} and we let T = {x: 3x – 10 = 34}. Clearly the test for membership in S is different from the test for membership in T. However, some elementary algebra shows us that both S and T are equal to {8}.

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© 2007 Herbert I. Gross next (1). The Empty (or, Null) Set Two Special Sets Sometimes the test for membership is so strict that nothing can survive it. For example, consider the set N = {x: x + 1 = x + 2}. It is relatively easy to see that x + 2 always exceeds x + 1 by 1. More specifically, x + 2 = (x +1) + 1. next

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© 2007 Herbert I. Gross next (1). The Empty (or, Null) Set So, as we did in Lessons 15 and 16, if we try to solve the equation x + 1 = x + 2 by subtracting x from both sides; we obtain the false statement 1 = 2. What this tells us is that if it were possible for x + 1 = x + 2, then it would mean that 1 = 2. Since this is a false statement, it tells us that x + 1 can never equal x + 2.

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© 2007 Herbert I. Gross next We give a special name to a set which has no members (e.g., A = {x:x+1 = x+2}). We call it the empty set or the null set, and we denote it by the symbol 0

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© 2007 Herbert I. Gross next At first glance one might wonder why we would even bother giving a name to a set which had no members. However, in terms of the set builder method for describing a set, the empty set is significant because it tells us that the test for membership is so strict that no number survives it. But, don't confuse the empty set with the set whose only member is 0 (that is {0}). For example, the set of all numbers that are neither positive nor negative is not empty. It consists of one number, namely 0.

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© 2007 Herbert I. Gross next (2). The Universal Set (Sometimes Referred to as the Universe of Discourse) At the other end of the spectrum from the empty set is the universal set which we may denote by u. The universal set contains every object that is eligible to be tested for membership. For example, it would make little sense to ask “Is the color blue a lawyer?”. That is, when we talk about the set of lawyers, only people are eligible to be members.

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© 2007 Herbert I. Gross next (2). The Universal Set In terms of our course: when we say that the only ways that 13 can be factored are 1 × 13 and 13 × 1, we are assuming that u, the universal set, is the set of whole numbers. Without this restriction, we could write such things as 13 = 1 / 2 × 26.

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© 2007 Herbert I. Gross next (2). The Universal Set As another example: when we say that the even numbers are 0, 2, 4, 6… we are assuming that the universal set is the set of non-negative integers (that is, the set of whole numbers). If the universal set had been the set of all integers, then the even numbers would have also included - 2, - 4, - 6…

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© 2007 Herbert I. Gross next The Universal Set and Identities If the solution set of an equation is the Universal Set ( the set of all numbers), it means the equation is an identity. That is: if the Universal Set is the solution set of an equation, it means that one side of the equation is a paraphrase of the other side. For example, the commutative property for addition tells us that {x: x + 1 = 1 + x} = u which is another way of saying that the equation x + 1 = 1 + x is an identity.

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© 2007 Herbert I. Gross next In part 2 of this lesson, we shall be looking at a more formal way to talk about formulas. That is, in the language of sets: a formula is a rule that assigns to a member of one set a member of another (or possibly the same) set. Looking Ahead

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