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**Algebra Problems… Solutions**

next Algebra Problems… Solutions Set 17 part 1 By Herbert I. Gross and Richard A. Medeiros © Herbert I. Gross

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**The set A has 8 members and However, the union of A and B**

next next Problem #1 The set A has 8 members and the set B has 5 members, However, the union of A and B (A U B) has only 10 members. How is this possible? Answer: A ∩ B has 3 members. © Herbert I. Gross 2

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**In terms of this exercise N(A) = 8,**

Answer: A ∩ B has 3 members. Solution for #1: Preliminary Notation We “invent” the notation N(A) to denote the number of members in the set A. Thus, N(B) denotes the number of members in B, N(A U B) denotes the number of members in A U B and N(A ∩ B) denotes the number of members in A ∩ B. next next In terms of this exercise N(A) = 8, N(B) = 5 and N(A U B) = 10. © Herbert I. Gross

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**counted once because it was **

Solution for # 1: If c is a member of both A and B, it was counted twice in computing the number of members in A U B. That is, it was counted once because it was a member of A and once because it was also a member of B. next next next In other words, all members in A ∩ B are counted twice. Hence, we must subtract the number of members in A ∩ B from the sum of the number of members in A and the number of members in B © Herbert I. Gross

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**N(A U B) = N(A) + N(B) – N(A ∩ B)**

Solution for #1: In more technical terms… next next next N(A U B) = N(A) + N(B) – N(A ∩ B) That is… 10 = – N(A ∩ B) 10 = N(A ∩ B) -3 = -N(A ∩ B) 3 = N(A ∩ B) © Herbert I. Gross

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next next Notes on #1 The formula… N(A U B) = N(A) + N(B) – N(A ∩ B) plays an important role in problems that involve being able to count accurately (such as in determining the probability of a particular outcome occurring). For example, in a standard deck of playing cards there are 13 spades and 12 face cards. © Herbert I. Gross 6

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**Suppose you want to know the cards and 13 spades; and 13 + 12 = 25.**

next next next Notes on #1 Suppose you want to know the number of ways that in a single draw from the deck you can obtain either a spade or a face card. Clearly, there are 12 face cards and 13 spades; and = 25. = 25 © Herbert I. Gross 7

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next next next Notes on #1 However, in arriving at 25, 3 cards were counted twice (namely the king, queen and jack of spades), Hence, there are only 22 ways in which you can obtain your objective. = 22 © Herbert I. Gross 8

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**two sets. In fact, while it might be N(A U B) = N(A) + N(B) – N(A ∩ B)**

next next Notes on #1 This exercise illustrates how one might tend to confuse adding with finding the number of members in the union of two sets. In fact, while it might be tempting to write… N(A U B) = N(A) + N(B) We see from the formula N(A U B) = N(A) + N(B) – N(A ∩ B) that this will be true only if N(A ∩ B) = 0 . © Herbert I. Gross 9

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**A Geometric Interpretation**

next next A Geometric Interpretation A geometric interpretation of the formula N(A U B) = N(A) + N(B) – N(A ∩ B) is known as a Venn diagram. Let the set A be represented by and let the set B be denoted by © Herbert I. Gross 10

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**A Geometric Interpretation**

next next next next next A Geometric Interpretation A U B is represented by the total area enclosed by the two rectangles, and A ∩ B is represented by the area of the region that is common to both rectangles (and is represented by ). 5 5 N(A) = 5 + 3 3 3 3 N(B) = 2 + 3 2 2 © Herbert I. Gross 11

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**A Geometric Interpretation**

next 5 2 3 In summary, we see from the diagram that there are 5 members of A that do not belong to B; there are 2 members of B that don’t belong to A; there are 3 members that belong to both A and B; and a total of 10 (that is ) members that belong to either A or B. © Herbert I. Gross 12

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next next Notes on #1 There is often a tendency to think of “Either..... or......” as meaning “One or the other, but not both”. However, the mathematical meaning is “At least one”. Thus, with respect to our earlier example of a spade or a face card”, if you were to pick the jack of spades you would still have won even though you picked both a spade and a face card. © Herbert I. Gross 13

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next next Notes on #1 Venn diagrams work nicely for two or three sets. However, they do not work if there are more than three sets. An alternative method involves using a table, in much the same way as we did when we wanted to record the outcomes of coins being flipped (Lesson 8). © Herbert I. Gross 14

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next next Notes on #1 For example, we may use 1 to indicate that a member belongs to a set and 0 to indicate that it doesn’t. So, for example, with respect to two sets (which we will denote by A and B) the table might look like… A B A U B A ∩ B 1 1 1 © Herbert I. Gross 15

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**This is where Boolean algebra is used.**

next next Notes on #1 However, just as in the case with flipping coins, as the number of sets increases the number of rows in our chart become rather unmanageable. This is where Boolean algebra is used. That is just as there are “rules of the game” in traditional algebra, there are rules in the “game” of Boolean algebra. Some of the rules are the same as they are in traditional algebra, but some are different as well. © Herbert I. Gross 16

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**The study of Boolean Algebra is very valuable but is beyond the**

next next Notes on #1 For example, the union of sets is commutative. That is, A U B = B U A. However, whereas a + a = 2a, A U A = A. An in depth study of unions and intersections is included in the course known as Boolean Algebra. The study of Boolean Algebra is very valuable but is beyond the scope of our course. © Herbert I. Gross 17

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**Answer: No Problem #2a Let A = {(x,y,z): x + y + z = 9}.**

next next Problem #2a Let A = {(x,y,z): x + y + z = 9}. Is (1,2,3) a member of A? Answer: No © Herbert I. Gross 18

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**pass the test for membership in the set A.**

Answer: No Solution for 2a: To belong to A, (x,y,z) must possess the property that… x + y + z = 9 next next next If we replace x by 1, y by 2 and z by 3 in the equation x + y + z = 9, we obtain the false statement… = 9 The fact that = 9 is a false statement means that (x,y,z} doesn’t pass the test for membership in the set A. © Herbert I. Gross

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**However, we are given the test for membership to belong to A.**

next next Notes on #2a As defined above, the set A is described implicitly. That is, we are not told specifically what the members of A are. However, we are given the test for membership to belong to A. This is the value of the set-builder notation. Namely it gives an objective criterion by which to determine whether a member belongs to it. © Herbert I. Gross 20

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**There are a number of ways to rewrite the equation x + y + z = 9.**

next next next Notes on #2a There are a number of ways to rewrite the equation x + y + z = 9. For example, the equation x + y + z = 9 is equivalent to the equation z = 9 – x – y. From the equation z = 9 – x – y, we see that if we choose the values of x and y in a completely arbitrary manner, the value of z is uniquely determined. © Herbert I. Gross 21

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next next Notes on #2a For example, if we let x = 5 and y = 3, then equation z = 9 – x – y tells us that… z = 9 – 5 – 3 = 1 Hence, (5,3,1) is a member of A. © Herbert I. Gross 22

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**in the equivalent form…**

next next next Notes on #2a In a similar way, we might have chosen to rewrite the equation x + y + z = 9 in the equivalent form… y = 9 – z – x In this case, we could choose x and z at random, and then the value of y would be uniquely determined by the equation y = 9 – z – x. © Herbert I. Gross 23

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next next Notes on #2a For example, if we let z = 2 and x = 4, then equation y = 9 – z – x tells us that… y = 9 – 2 – 4 = 3 Hence, (4,3,2) is a member of A. © Herbert I. Gross 24

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**Answer: Yes Problem #2b Let A = {(x,y,z): x + y + z = 9}.**

next next Problem #2b Let A = {(x,y,z): x + y + z = 9}. Is (1,2,6) a member of A? Answer: Yes © Herbert I. Gross 25

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**To belong to A, (x,y,z) must possess the property that… x + y + z = 9**

Answer: Yes Solution for #2b: To belong to A, (x,y,z) must possess the property that… x + y + z = 9 next next next If we replace x by 1, y by 2 and z by 6 in the equation above we obtain the true statement… = 9 The fact that = 6 is a true statement means that (1,2,3} satisfies the test for membership in the set A © Herbert I. Gross

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**x + y + z = 9 tells us that z must equal 6.**

next next Notes on #2b Notice that in both parts of this exercise the value of x was 1 and the value of y was 2. However (1,2,3) didn’t belong to set A, but (1,2,6) does. The reason for this is that once we know that x = 1 and y = 2, the equation x + y + z = 9 tells us that z must equal 6. z = 9 = 9 © Herbert I. Gross

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**set A tells us that since (1,2,6) is a member**

next next Notes on #2b Because addition is associative and commutative, the test for membership in set A tells us that since (1,2,6) is a member of set A then so also are (1,6,2), (2,1,6), (2,6,1), (6,1,2) and (6,2,1). That is… = 9 = 9 = 9 = 9 = 9 = 9 © Herbert I. Gross

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**How are sets A and B related if**

next next Problem #3 How are sets A and B related if A = {1,2,3} and B = {x:(x – 1)(x – 2)(x – 3) = 0}? Answer: They are equal. © Herbert I. Gross 29

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**B are, but we are told the test for membership in B.**

Answer: They are equal. Solution for #3: The set A is described by the roster method. That is, its members are explicitly listed. That is, for a number to belong to A, it must be either 1, 2 or 3. next next On the other hand the set builder notation is used to describe set B. That is, we are not told explicitly what the members of B are, but we are told the test for membership in B. © Herbert I. Gross

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**Namely for a number x to belong to B it must satisfy the equation…**

Solution for #3: Namely for a number x to belong to B it must satisfy the equation… (x – 1)(x – 2)(x – 3) = 0 next next Notice that the equation… (x – 1)(x – 2)(x – 3) = 0 tells us that every member of A is also a member of B. © Herbert I. Gross

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**For example, since the product of any number and 0 is 0, we see that…**

Solution for #3: For example, since the product of any number and 0 is 0, we see that… next next If x = 1, x – 1 = 0; and if x – 1 = 0, (x – 1)(x – 2)(x – 3) = 0 If x = 2, x – 2 = 0; and if x – 2 = 0, (x – 1)(x – 2)(x – 3) = 0 If x = 3, x – 3 = 0; and if x – 3 = 0, (x – 1)(x – 2)(x – 3) = 0 © Herbert I. Gross

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**Thus, with respect to the equation (x – 1)(x – 2)(x – 3) = 0, **

Solution for #3: On the other hand, the only way a product can equal 0 is if at least one of its factors is equal to 0. next next next Thus, with respect to the equation (x – 1)(x – 2)(x – 3) = 0, the only way the product can equal 0 is if either (x – 1) = 0, (x – 2) = 0 or (x – 3) = 0; that is only if x = 1, x = 2, or x = 3. Hence, in the roster method format B = {1,2,3}. Thus the sets A and B are, in effect, two different names for same set. © Herbert I. Gross

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**an equation from the set builder notation to the roster notation.**

next next Notes on #3 In terms of implicit versus explicit, we may say that part of the mission of algebra is to help us convert the solution set of an equation from the set builder notation to the roster notation. For example, in terms of this exercise, if we define the set B by {x:(x – 1)(x – 2)(x – 3) = 0} then algebra is the process of showing that B = {1,2,3}. © Herbert I. Gross

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next next Notes on #3 In general we’re more comfortable seeing the solution set in its roster format. However the set-builder notation tells us the test for membership. For example, in a previous problem, we were looking at the set… A = {(x,y,z): x + y + z = 9}. In this case, it is impossible to list all the members of A because there are infinitely many such members (one for each choice of x and y). © Herbert I. Gross

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next next Notes on #3 However, as we saw in the solution of the exercise, we could use the test for membership to determine whether a given “triplet” of numbers (x,y,z) belonged to the set A. As another example, it is known that there are infinitely many prime numbers (recall that a prime number is any whole number greater than 1 that is divisible only by 1 and itself). © Herbert I. Gross

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next next Notes on #3 However, there is no known formula for listing them all. But, given any whole number greater than 1 we can test to see if it has any divisors other than 1 and itself. For example, given the number 1,234,576,926 we see at once that it is even and hence divisible by 2. Therefore, it doesn’t belong to the set of prime numbers. © Herbert I. Gross

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**roster representation. For example, when we see {1,2,3} we**

next next Final Note on #3 As a final note, observe that in a very important way the set builder notation tells us things we might not observe from the roster representation. For example, when we see {1,2,3} we do not know the sense in which these three numbers were chosen for membership. For example, we might have chosen them because they were the first three positive integers. However, when we see {x:(x – 1)(x – 2)(x – 3) = 0} we know immediately what the test for membership was. © Herbert I. Gross

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**Answer: It is the set of all multiples of 6.**

next next Problem #4 Let A be the set of positive integers that are multiples of 3 and let B be the set of positive integers that are multiples of 2. Describe the set A ∩ B. Answer: It is the set of all multiples of 6. © Herbert I. Gross 39

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**Answer: It is the set of all multiples of 6. Solution for #4: **

Using the roster method we see that A = {3,6,9,12,15,..., 3n,...}. Because the set A has infinitely many members, we might prefer to write it in the set builder notation; namely… A = {3n:n is a positive integer} next next The key point is that to be member of A, a positive integer has to be divisible by 3. © Herbert I. Gross

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**In a similar way, we may represent B in the form… **

Solution for #4: In a similar way, we may represent B in the form… B = {2n:n is a positive integer} next next The key point is that to be member of B, a positive integer has to be divisible by 2. © Herbert I. Gross

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**Thus, to belong to both A and B (that is, to A ∩ B) the positive **

Solution for #4: Thus, to belong to both A and B (that is, to A ∩ B) the positive integer has to be divisible by both 2 and 3. Any number that is divisible by both 2 and 3 is also divisible by 6; and conversely, any number divisible by 6 is also divisible by both 2 and 3. next next Hence, A ∩ B = {6n:n is a positive integer}; or in roster format… A ∩ B = {6,12,18,..., 6n, ...} © Herbert I. Gross

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**For example, 4 × 6 is divisible**

next next Notes on #4 In our solution to this exercise, we used the fact that 6 was the least common multiple of 2 and 3. It is always true that the product of two numbers is divisible by each of the two numbers. For example, 4 × 6 is divisible by both 4 and 6. © Herbert I. Gross

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**The reason for this is that 4 and 6 share the factor 2 in common.**

next next Notes on #4 However, in the case of 4 and 6, 4 × 6 is not the least common multiple of 4 and 6. Rather 12 is the least common multiple of 4 and 6. The reason for this is that 4 and 6 share the factor 2 in common. That is 4 = 2 × 2 and 6 = 2 × 3. © Herbert I. Gross

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**So in terms of a diagram…**

next next next next next Notes on #4 So in terms of a diagram… 4 2 2 4 = 2 × 2 2 2 2 6 = 3 × 2 3 3 6 The diagram shows that to be divisible by both 4 and 6 it has to be a multiple of 2 × 2 × 3 © Herbert I. Gross

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**Answer: A ∩ B = {(3,19)} Problem #5 Let A = {(x,y):y = 4x + 7} and**

next next Problem #5 Let A = {(x,y):y = 4x + 7} and let B = {(x,y):y = 2x + 13}. Describe the set A ∩ B. Answer: A ∩ B = {(3,19)} © Herbert I. Gross 46

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**In set builder notation we know that… **

Answer: A ∩ B = {(3,19)} Solution for #5: In set builder notation we know that… A ∩ B = {(x,y):y = 4x + 7and y = 2x + 13}. next next The only way in which y can be equal to both 4x + 7 and 2x + 13 is if… 4x + 7 = 2x + 13 © Herbert I. Gross

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**To solve the equation 4x + 7 = 2x + 13 for **

Solution for #5: To solve the equation 4x + 7 = 2x + 13 for x, we may first subtract 2x from both sides of the equation to obtain… 2x + 7 = 13 next next next We then subtract 7 from both sides of the equation 2x + 7 = 13 to obtain… 2x = 6 …and from the equation 2x = 6, it follows that x = 3. That is, if x = 3 then 4x + 7 = 2x + 13 = 19. © Herbert I. Gross

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next next Notes on #5 Even if we didn’t know how to use algebra to convert from the set builder notation to the roster method, we could still use the test for membership. For example, suppose we wanted to see whether (5,27) is a member of A∩B. © Herbert I. Gross

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next next Notes on #5 If we replace x by 5 and y by 27 in the equation y = 4x + 7, we obtain the true statement… 27 = 4(5) + 7. However, if we replace x by 5 and y by 27 in the equation y = 2x + 13, we obtain the false statement… 27 = 2(5) + 13 © Herbert I. Gross

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**to B. Hence, (5, 27) is not a member of the set A ∩ B.**

next next Notes on #5 In other words (5,27) belongs to A but not to B. Hence, (5, 27) is not a member of the set A ∩ B. The beauty of the algebraic solution is that it eliminates the need to test other pairs of numbers. Namely, the algebraic solution told us that (x,y) was a member of the set A ∩ B if and only if x = 3 and y = 19. © Herbert I. Gross

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