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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 3

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

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Graphing Revisited In our course we have been emphasizing the concepts of direct and indirect computations. In the language of functions this may be summarized as follows… next Suppose we are given the function f, where… f(x)= 3x + 2. © 2007 Herbert I. Gross

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Direct computation is when we are given the value of x and are told to find the corresponding value of f(x). So, for example, if x = 4, f(x) = f(4) = 3(4) + 2 = 14. Indirect computation is when we are given the value of f(x) and are told to find the corresponding value of x. So, for example, if f(x) = 14, x = 4. next © 2007 Herbert I. Gross

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Guess The Function However, a third possibility (but one we haven’t discussed as yet) is when we are given the values of x and the corresponding values of f(x); and we are asked to “guess” what rule is defined by the function f. It is this scenario that might lead one to “discover” graphing. That is, we could locate the points (x, f(x)) in the xy-plane and see if they seemed to form a “familiar” curve. next © 2007 Herbert I. Gross

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To Summarize (1) Arithmetic is when we are given x and f and we are asked to find f(x); that is, we are given the input and the function and asked to find the output. (2) Algebra is when we are given f(x) and f and we are asked to find x; that is, we are given the output and the function and asked to find the input. (3) Geometry (more specifically, graphing) is when we are given x and f(x) and asked to determine f. next © 2007 Herbert I. Gross

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Mathematics Versus the “Real World” © 2007 Herbert I. Gross next So far we have dealt with either situation (1) or situation (2) but have not had to deal with using the values of x and f(x) to determine f. However in the “real world” scientists usually try to “discover” a law of nature by making various inputs and then observing the corresponding outputs.

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From Graphs to Functions © 2007 Herbert I. Gross next More specifically, in the real world we try to discover relationships by studying patterns. In this context we make an input and measure the corresponding output. Using the x-axis to denote the input and the y-axis to denote the output we graph the set of points (input, output). We then look at the resulting set of points and try to visualize the most likely curve that would pass through these points. We represent this curve by y = f(x)

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© 2007 Herbert I. Gross next For example, suppose… Input Output Input O u t p u t next y = f(x)

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Mathematics Versus the “Real World” © 2007 Herbert I. Gross next This process leads to what we call inductive reasoning, as opposed to the deductive reasoning we use in mathematics.

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The Case Against Patterns… Inductive Versus Deductive © 2007 Herbert I. Gross next As an introduction to inductive reasoning consider the following sequence of numbers: 31, 30, 31, 30, 31, 31 30, 31, 30, 31, 31,... The sequence obeys a well-defined rule. Based on this, can you guess whether the next number in the sequence is 30 or 31?

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© 2007 Herbert I. Gross next While there are many good rationales for defending either of the two choices, the answer is that the next number is 28. Namely, the pattern that the above sequence was following was to list the number of days in a month in a non leap year, starting with March. 31, 30, 31, 30, 31, 31 30, 31, 30, 31, 31,... M a r c h A p r i l M a y J u n e J u l y A u g u s t S e p t e m b e r O c t o b e r N o v e m b e r D e c e m b e r J a n u a r y F e b r u a r y next

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© 2007 Herbert I. Gross next The point is that when we were given the above sequence of numbers we had to draw a conclusion based on the available evidence. The best we could say about our conclusion was that it was plausible. However, once we knew explicitly what the rule was, we could deduce that the next term had to be 28 and that our conclusion was incorrect.

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© 2007 Herbert I. Gross next Applying this to our previous example… Let’s guess the value of f(5.5) Input OutputOutput y = f(x) 5.5 f(5.5) = 4.5 To validate our guess, we would input 5.5 and see how close the actual output was to 4.5. next

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© 2007 Herbert I. Gross next The drawback to this approach is that as long as there are spaces between the points we have drawn; it is only a guess as to what the curve actually looks like. For example: or… or… next or ? ? ? ?…

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© 2007 Herbert I. Gross next When the best we can do is to make an “educated guess” we say that we obtained the conclusion inductively. When we know that the conclusion follows inescapably from the given information we say that we obtained the conclusion deductively. The two forms of reasoning often occur side by side.

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© 2007 Herbert I. Gross next For example, consider the following logical argument… Assumption 1: All dogs bark. Assumption 2: Rover is a dog. Conclusion: Therefore, Rover barks. The conclusion is said to have been obtained by deductive reasoning; meaning that it followed inescapably from the given assumptions. And in such a case we say that the argument is valid

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© 2007 Herbert I. Gross next However, the assumption that all dogs bark was based on what we call inductive or empirical evidence. That is, we certainly did not test every dog in the world in order to arrive at Assumption 1. Rather based on what we were able to observe we assumed that all dogs bark. The subtle point here is that even if it's false that all dogs bark, the above argument is still valid. Namely, if all dogs bark and if Rover is a dog, then Rover barks.”

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© 2007 Herbert I. Gross next There are situations in which the conclusion is true, but the argument is not valid. For example… Assumption 1: All dogs bark. Assumption 2: Rover barks. Conclusion: Therefore, Rover is a dog. In this case the argument is not valid. The conclusion might be true but the truth doesn't follow inescapably from the assumptions. For example, it's possible that other animals (such as seals) also bark.

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© 2007 Herbert I. Gross next Notice how subtle logic is. For example, in the above argument if we change Assumption 1 to read “Only dogs bark”, then the argument “If Rover barks, Rover is a dog” is valid. Note In summary, truth is associated with a statement and validity is associated with an argument. The connection between the two is that if the assumptions are true and the argument is valid, then the conclusion will also be true. next

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© 2007 Herbert I. Gross next A Valid Argument and a False Conclusion. In this case the conclusion is false, but it follows inescapably from the assumptions (at least one of which is false). Illustrative Example Assumption 1: All dogs have 5 legs. Assumption 2: Rover is a dog. Conclusion: Therefore, Rover has 5 legs. next

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© 2007 Herbert I. Gross next An Invalid (that is, not valid) Argument and a True Conclusion. In this case the conclusion happens to be true but it doesn't follow from the assumptions. Illustrative Example A1: All Parisians are Europeans. A2: All Frenchmen are Europeans. C: Therefore all Parisians are Frenchmen. next

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© 2007 Herbert I. Gross next One way to see this is to replace “Parisians” by “Germans”. In this case, the form of the argument remains unchanged but the argument would become… Illustrative Example A1: All Parisians are Europeans. A2: All Frenchmen are Europeans. C: Therefore, all Parisians are Frenchmen. next Germans Germans

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© 2007 Herbert I. Gross next To see how the above discussion differs from what is emphasized in our course, let's suppose that we are measuring inputs and the corresponding outputs, and we find that when the input is 0 the output is 0, and when the input is 2 the output is 4 xf(x) or in the language of functions… That is… next And, further, suppose that this was the only information we had, from which we had to inductively construct the function, f. inputoutput 00 24

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© 2007 Herbert I. Gross next If we knew that f was linear, we could deduce that f(x) = 2x. However, in the above situation we are not told that this is the case. In fact suppose that f was defined by f(x) = x 2. In this case it would also be true that f(0) = 0 and f(2) = 4. In summary, the important point is that when we use inductive reasoning we can never be sure what will happen next until after it happens.

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© 2007 Herbert I. Gross next In our course however, we have always been told what f was; after which we were either called upon to determine f(x) once x was known (direct computation) or to determine x once f(x) was known (indirect computation). With respect to our course we would have been told the rule.

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© 2007 Herbert I. Gross next For example: We might have been told that f is defined by f(x) = 2x; in which case we could have logically deduced that f(3) = 2(3) = 6, next or we might have been told that f is defined by f(x) = x 2 ; in which case we could have logically deduced that f(3) = (3) 2 = 9.

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© 2007 Herbert I. Gross next Review When we first defined graphs, we said that the graph of the function f is the set of all ordered pairs (x,f(x)). This definition transcends the graph being restricted to sets of numbers. next For example, in part 2 of this lesson, we talked about the grading function. By way of review, recall that…

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© 2007 Herbert I. Gross next Review next And the function f was the teacher who assigned the grade A- to Tom, B+ to Bill, and A to Jane. The set S was the set consisting of the students Tom, Bill, and Jane. The set G was the set that consisted of the grades A-, B+, and A.

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© 2007 Herbert I. Gross next In summary… Tom Bill Jane A- B+ A f f f next G S and in the language of functions we see that… f(Tom) = A- f(Bill) = B+ f(Jane) = A

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(Tom, A-), (Bill, B+), and (Jane, A) And this can be represented geometrically as follows… Thus the graph of f is the set of ordered pairs… © 2007 Herbert I. Gross

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B+ A- A Tom Bill Jane G S next © 2007 Herbert I. Gross (Bill, B+) (Tom, A) (Jane, A-)

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© 2007 Herbert I. Gross next However, the important point is to always keep in mind that every function has a graph and the graph exists without our having to represent it pictorially. Key Point

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More on Using Graphs to Solve Algebraic Equations © 2007 Herbert I. Gross next As an introduction to inductive reasoning, consider that up until now in our course there is really no need to use graphing to help us solve algebraic equations. Namely, the only equations we have encountered so far can be written in the form… mx + b = nx + c.

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© 2007 Herbert I. Gross next In such cases, we simply rewrite the equation in a form where only terms involving x are on one side and the terms that don't involve x (that is, the constant terms) are on the other side. By way of review let’s solve the equation… 5x + 3 = 3x + 17 mx + b = nx + c.

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© 2007 Herbert I. Gross next We may subtract 3x from both sides of the equation 5x + 3 = 3x + 17 to obtain the equivalent equation… 2x + 3 = 17. We may then subtract 3 from both sides of this equation to obtain… 2x = 14. Then dividing by 2, we see that the solution is… x = 7. As a check notice that when x = 7, both 5x + 3 and 3x + 17 are equal to 38.

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© 2007 Herbert I. Gross next However, not all algebraic equations are this easy to solve. For example, consider an equation such as… x 2 = 2x + 3 next In terms of our rules for addition, = 3 whenever 1, 2, and 3 modify the same noun. Hence, 1x + 2x = 3x and 1x 2 + 2x 2 = 3x 2. However in an expression such as 1x 2 + 2x, we cannot combine the terms because x 2 and x are different nouns.

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© 2007 Herbert I. Gross next In other words in the equation, x 2 = 2x + 3, it doesn't help us if we subtract 2x from both sides because there will be two terms, both of which contain x but cannot be combined into a single term. In a subsequent course, we will show how to solve such an equation algebraically, but for now we want to show you how it helps to draw the graphs of y = 2x + 3 and y = x 2. Namely the equation x 2 = 2x + 3 represents the x-coordinates of the points at which the curves y = x 2 and y = 2x + 3 intersect.

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© 2007 Herbert I. Gross next Therefore one way to proceed is as follows… (1) Draw the curve (in this case a straight line) whose equation is y = 2x + 3. One way to do this is to use the fact that b = 3 tells us that the point (0,3) is on the line; and the fact that m = 2 tells us that the line rises by 2 units for every unit it moves to the right.

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© 2007 Herbert I. Gross next (0,0) y = 2x (0,3) 2

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© 2007 Herbert I. Gross next (2) The curve whose equation is y = x 2 is not linear. Hence, it is not as easy to sketch. What we can do is locate several points that belong to the curve and then use induction to guess what the curve looks like. For example, from the table: xx2x2 y = x /21/2 1/41/4 1/41/

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© 2007 Herbert I. Gross next We see that the curve whose equation is y = x 2 contains, among others, the points… (0,0), (1,1), (2,4), (3,9), ( 1 / 2, 1 / 4 ) and ( - 2, 4)

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next © 2007 Herbert I. Gross next (0,0) y = x 2 In terms of a graph… (1,1) (2,4) (3,9) (-1,1) ( - 2,4) ( - 3,9)

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next © 2007 Herbert I. Gross next (0,0) y = 2x + 3 (3,9) ( - 1,1) y = x 2 Sketched in the same grid…

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© 2007 Herbert I. Gross next From this diagram, we see that the two curves appear to intersect at the points (3,9) and ( - 1,1). We may check that x = 3 and x = - 1 are actually solutions of the equation. x 2 = 2x + 3 next In particular, if we replace x by 3 in this equation, we obtain the true statement… (3) 2 = 2(3) + 3(= 9) And if we replace x by - 1 in this equation, we obtain the true statement… ( - 1) 2 = 2( - 1) + 3 (= 1) next

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© 2007 Herbert I. Gross next We could have used a chart instead of the geometric graph, but we would still have had to contend with the rather subtle question… Note How do we know that we have found all the values of x that satisfy the equation x 2 = 2x + 3 ? next

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© 2007 Herbert I. Gross next For example, if when we made the table we looked only at positive values of x, we would have missed the solution x = - 1. And even if we did find that x = - 1 was a solution; how would we know that there weren't other solutions that we have overlooked? However, if we assume that we have correctly guessed the geometric graph of y = x 2 the diagram shows us that these are the only two points of intersection that the two curves possess.

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© 2007 Herbert I. Gross next While it is beyond the scope of this course to dwell on functions of more than one variable, the fact is that in most real life situations it is rare to encounter situations where the outcome is dependent on a single input (variable). Note on Functions with More Than One Variable For example, in some of our earlier examples we were concerned with buying… pens that cost $3 per box or candy that cost $4 per box.

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© 2007 Herbert I. Gross next However, when we go to a store we often buy more than one type of item. Thus, for example, it's possible that we might want to buy both the pens and the candy. So suppose we were buying x boxes of pens and y boxes of candy. -- The cost of the candy (in dollars) would be… 4y -- The cost of the pens (in dollars) would be… 3x and

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© 2007 Herbert I. Gross next Hence, the total cost T (in dollars) would be given by the formula… T = 3x + 4y …which in the language of functions might be expressed as… T = f(x,y) where f(x,y) = 3x + 4y

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© 2007 Herbert I. Gross next While it is common to use x and y as generic names for the variables, in specific examples it is often helpful to choose more suggestive letters, such as p to represent the number of boxes of pens and c to represent the number of boxes of candy. Note

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© 2007 Herbert I. Gross next In science and technology, as an abbreviation for… T(x,y) = 3x + 4y …one often simply writes… T = f(x,y) where f(x,y) = 3x + 4y Caution

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© 2007 Herbert I. Gross next Unless one is very comfortable with mathematical usage this can cause confusion because in this case T is being used to represent two different things. (1) It represents the total cost of the pens and the candy. Hence, even though it takes a bit longer, at least at the beginning, we prefer to write the relationship in the form… T = f(x,y), where f(x,y) = 3x + 4y (2) It represents the “recipe” for finding the total cost.

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© 2007 Herbert I. Gross next The graph of f is the set of all 3-tuples (ordered triplets)… Graphing (where x and y are restricted to being whole numbers if we assume that one can only buy a whole number of boxes). (x,y,T) = (x,y,3x + 4y)

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© 2007 Herbert I. Gross next If we were to draw the graph, we would need the xy-plane to represent the set of inputs (x,y); and a third dimension (which is usually represented by a number line, called the z-axis, that is perpendicular to the xy-plane). Graphing (x,y,z) next In this case the graph would be a set of points that constituted a surface (rather than a curve) in the xyz-coordinate system.

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© 2007 Herbert I. Gross next While the geometric graph now requires 3-dimensions, it is still possible to draw it. Graphing (x,y,z) next However, if we were to buy as few as three different items, we would not be able to draw the graph of the resulting function even though the graph still exists algebraically.

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© 2007 Herbert I. Gross next For example, suppose that in addition to buying boxes of pens at $3 per box and boxes of candy at $4 per box, we were also buying greeting cards at $2 per card. In this case the total cost (T in dollars) of buying x boxes of pens, y boxes of candy, and z greeting cards would be given by… Graphing (x,y,z) next T = g(x,y,z), where g(x,y,z) = 3x + 4y + 2z

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© 2007 Herbert I. Gross next The graph of g is the set of 4-tuples (x, y, z, 3x + 4y + 2z) where x, y and z are whole numbers. Mathematically speaking, the graph is a 4-dimensional space; but, geometrically, it would require a 4-dimensional space to draw it! Graphing 4-tuples next So from a mathematical point of view, it is just as logical to talk about 4-dimensional space as it is to talk about a 1-, 2-, or 3- dimensional space. More specifically, mathematicians identify an n-dimensional space with the arithmetic of n-tuples.

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© 2007 Herbert I. Gross next This completes our current discussion of graphs. In the next three lessons we will look at functions in greater detail. Closing Comments next In particular, in Lesson 18 we will look at the arithmetic of linear functions in more depth; in Lesson 19 we will look further at exponential functions; and in Lesson 20 we will discuss inverse functions in more depth.

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