......!!!!!!!!...................................
09:26:45 Query 2.1.12 counting #'s 4 to 14
List the elements of the set.......!!!!!!!!...................................
RESPONSE --> 5,6,7,8,9,10,11,12,13 - counting numbers between 4 and 14
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09:28:19 **A list of the elements would just be 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. **
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RESPONSE --> I included all numbers instead of 4 and 14. I gave this answer because I assumed it meant to list the numbers 'between' 4 and 14.
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09:29:41 query 2.1.24 set builder for set of presidents between LBJ and Clinton
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RESPONSE --> {Nixon, Ford, Carter, Reagan, Bush} Set of presidents that served between LBJ and Clinton.
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09:40:53 ** Set-builder notation is {x|x is a president who served between Lyndon Johnson and William Clinton}
x is a variable and the condition 'x is a president who served between Lyndon Johnson and William Clinton' tells you what possible things the variable can be. COMMON ERROR: It's incorrect to say {x | x is the set of presidents who served between Johnson and Clinton}. x is a president, not a set of presidents. Should be {x|x is a president who served between Lyndon Johnson and William Clinton} **......!!!!!!!!...................................
RESPONSE --> Ok, I'm a little confused because the answer I gave to this question is answered like the other questions from its section in the book based on the answers in the back. I do understand the answer that you have given and how you came to get it but I think either I misunderstood the question or you asked a different question than the book.
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09:43:51 2.1.40 finite or infinite: set of rat #'s 0 to 1
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RESPONSE --> infinite-because there is an infinate number of cardinal numbers between 0 and 1 (ex: 0.001, 0.002, 0.003, ...)
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09:44:16 ** Rational numbers have form p/q, where p and q are integers. Numbers like 5/8, 57/31, -3/5, -57843/7843, etc.
The subset {1/2, 1/3, 1/4, 1/5, ... } is just by itself an infinite set of rational numbers between 0 and 1. Then you have things like 348/937, and 39827389871 / 4982743789, and a whole infinite bunch of others. There are thus infinitely many rational numbers in any interval of the real line. COMMON MISCONCEPTION: finite, because it doesn't go on forever Rational numbers have form p/q, where p and q are integers. Numbers like 5/8, 57/31, -3/5, -57843/7843, etc. Not all of these lie between 0 and 1, of course. **......!!!!!!!!...................................
RESPONSE --> OK
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09:52:39 2.1.48 n(A), A={x|x is a U.S. senator}
What is n(A) and why?......!!!!!!!!...................................
RESPONSE --> n(A) = {George Allen}
This is because the question asked A={xlx us a U.S. senator} and this means that it is asking the answer to be given in set builder notation..................................................
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09:56:33 ** n(A) stands for the number of elements in the set--in this case for the number of senators. There are 100, 2 from each State. So n(A) = 100. **
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RESPONSE --> Ok, I went back and read page 51 of the textbook. This page says that n(A) represents the cardinal number of a set. I should have included the total number of US senators instead of listing one.
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10:01:53 query 2.1.54 {x|x is neagtive number}
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RESPONSE --> This is well defined because it says that x is a negative number and this means that any negative number belongs in the set.
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10:02:14 ** This notation means all possible values of x such that x is a negative number.
The question is whether the set is well-defined or not. It is in fact well-defined because there is a definite way to decide whether a given object is an element of the set, because there is a definite way to determine whether an object is a negative number or not. ALTERNATIVE ANSWER: The set is well-defined because you have a criterion by which you can definitely decide whether something is or is not in the set. **......!!!!!!!!...................................
RESPONSE --> OK
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10:03:19 2.1.72 true or false 2 not subset of {7,6,5,4}
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RESPONSE --> false because 2 is not included in the set of {7,6,5,4}
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10:05:51 ** The statement is that 2 is not a subset. The statement is true because 2 isn't even a set, it's just a number. {2} is a set and could be a subset of something. 2 is just a number; it isn't a set so it can't be a subset of anything.
The usual answer is that 2 isn't a subset because 2 isn't in the set. However that's not the correct reason. The correct reason is that 2 isn't a set and a subset must be a set. COMMON MISCONCEPTION: the statement says that 2 is not a subset, not that it is not an element of the set. So the reason it's not a subset is that 2 isn't a set at all, so it can't be a subset of anything. **......!!!!!!!!...................................
RESPONSE --> OK, I understand why I missed this question. The 2 is not a set, it is only a number. If it were a set, it would be given as {2}, which is not a subset of the given set.
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10:07:18 2.1.84 C={4,10,12}, B={2,4,8,10}
Is it true or false that every element of C is also an element of B?......!!!!!!!!...................................
RESPONSE --> False- if every element of C were also an element of B, B would have to include 12.
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10:07:25 ** Specifically it's false because the element 12 is in C but not in B. **
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RESPONSE --> OK
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vĩݛL assignment #002 g}ၳQ} Liberal Arts Mathematics I 09-11-2005
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10:09:29 2.2.24 U={a,b,...,g}, A={a,e}, B={a,b,e,f,g}, C={b,f,g}, D={d,e} Is C ps U?
Is the statement true or false and why?......!!!!!!!!...................................
RESPONSE --> True-because every element of C is also in U.
.................................................
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10:09:39 ** It's true because all elements of C are in the universal set, and because there are elements of U that aren't in C. You have to have both conditions, since a proper subset cannot be identical to other set. **
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RESPONSE --> OK
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10:12:25 Query 2.2.30 phi s D
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RESPONSE --> False-because 0 is an empty set so it could be a proper set of D.
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10:13:42 ** Note that You should be responding to problem 2.2.30 from the homework you worked out on paper. The shorthand notation is for my reference and won't always make sense to you. For clarification, though, the symbol for the empty set is the Greek letter phi.
One set is a subset of another if every element of that set is in the other. To show that a set isn't a subset of another you have to show something in that set that isn't in the other. There's nothing in the empty set so you can never do this--you can never show that it has something the other set doesn't. So you can never say that the empty set isn't a subset of another set. Thus the empty set is a subset of any given set, and in particular it's a subset of D. ALTERNATIVE ANSWER: As the text tells you, the empty set is a subset of every set. ANOTHER ALTERNATIVE Every element of the empty set is in D because there is no element in the empty set available to lie outside of D. ONE MORE ALTERNATIVE: The empty set is a subset of every set. Any element in an empty set is in any set, since there's nothing in the empty set to contradict that statement. **......!!!!!!!!...................................
RESPONSE --> OK
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10:14:51 2.2.33 D not s B
Is the statement true or false and why?......!!!!!!!!...................................
RESPONSE --> True-because set D contains the letter d and set B does not.
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10:15:11 ** D is a subset of B if every element of D is an element of B-i.e., if D doesn't contain anything that B doesn't also contain.
The statement says that D is not a subset of B. This will be so if D contains at least one element that B doesn't. **......!!!!!!!!...................................
RESPONSE --> OK
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10:17:31 2.2.36 there are exactly 31 subsets of B
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RESPONSE --> False-because there are 5 elements in B which means that you take 2 to the power of 5(elements in the set) to get the number of subsets.
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10:18:00 ** If a set has n elements then is has 2^n subsets, all but one of which are proper subsets. B has 5 elements so it has 2^5 = 32 subsets. So the statement is false.
There are exactly 31 proper subsets of B, but there are 32 subsets of B. **......!!!!!!!!...................................
RESPONSE --> OK.
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10:19:49 Query 2.2.40 there are exactly 127 proper subsets of U
Is the statement true or false and why?......!!!!!!!!...................................
RESPONSE --> False-because to find proper subsets, you have to take the number 2 and take to the power of the number of elements in the set (7) and this gives you 128. Then you have to subtract 1 to get the number of proper subsets.
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10:20:01 ** The set is not a proper subset of itself, and the set itself is contained in the 2^n = 2^7 = 128 subsets of this 7-element set. This leaves 128-1 = 127 proper subsets. **
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RESPONSE --> OK
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10:24:34 Query 2.2.48 U={1,2,...,10}, complement of {2,5,7,9,10}
What is the complement of the given set?......!!!!!!!!...................................
RESPONSE --> Ok, question 48 in the book does not ask this question so I'll answer the one you have given here. This is actuallly question 50 in our book.
{1,3,4,6,8}.................................................
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10:24:51 ** the complement is {1,3,4,6,8}, the set of all elements in U that aren't in the given set. **
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RESPONSE --> OK, I got this one right.
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10:28:39 query 2.2.63 in how many ways can 3 of the five people A, B, C, D, E gather in a suite?
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RESPONSE --> There are 10 ways. {A,B,C} {A,B,D} {A,B,E} {A,C,D} {A,C,E} {A,D,E} {B,C,D} {B,C,E} {B,D,E} {C,D,E}
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10:28:53 ** The answer here would consist of a list of all 3-element subsets: {a,b,c}, {a,b,d}, {a,b,e}, {a,c,d} etc. There are ten such subsets.
Using a,b,c,d,e to stand for the names, we can list them in alphabetical order: {a,b,c), {a,b,d}, {a,b,e}, {a,c,d}, {a,c,e}, {a,d,e|, {b,c,d}, {b,c,e}, {b,d,e}, {c, d, e}**......!!!!!!!!...................................
RESPONSE --> OK
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صBO_걈 Student Name: assignment #004 004. Liberal Arts Mathematics
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10:36:03 `q001. Consider the statement 'If that group of six-year-olds doesn't have adult supervision, they won't act in an orderly manner.' Under which of the following circumstances would everyone have to agree that the statement is false?
The group does have supervision and they do act in an orderly manner. The group doesn't have supervision and they don't act in an orderly manner. The group doesn't have supervision and they do act in an orderly manner. The group does have supervision and they don't act in an orderly manner.......!!!!!!!!...................................
RESPONSE --> The group doesn't have supervision and they do act in an orderly manner.
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10:36:33 The statement says that if the group doesn't have supervision, they will not act in an orderly manner. So if they don't have supervision and yet do act in an orderly manner the statement is contradicted.
If the group does have supervision, the statement cannot be contradicted because condition of the statement, that the group doesn't have supervision, does not hold. The statement has nothing to say about what happens if the group does have supervision. Of course if the group doesn't have supervision and doesn't act in orderly manner this is completely consistent with the statement. Therefore the only way to statement can be considered false is the group doesn't have supervision and does act in an overly manner. Note that what we know, or think we know, about childrens' behavior has nothing at all to do with the logic of the situation. We could analyze the logic of a statement like 'If the Moon is made of green cheese then most six-year-olds prefer collard greens to chocolate ice cream'. Anything we know about the composition of the Moon or the tastes of children has nothing to do with the fact that the only way this statement could be shown false would be for the Moon to be made of green cheese and most six-year-olds to prefer the ice cream.......!!!!!!!!...................................
RESPONSE --> OK
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10:38:37 `q002. List the different orders in which the letters a, b and c could be arranged (examples are 'acb' and 'cba'). Explain how you know that your list contains every possible order.
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RESPONSE --> abc bac cab bca acb cba
My list contains every possible order because each letter gets a chance to be first second and third in the list..................................................
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10:39:09 The only reliable way to get all possible orders is to have a system which we are sure the list every order without missing any.
Perhaps the simplest way to construct all possible orders is to list then alphabetically. We start with abc. There is only one other order that starts with a, and it is obtained by switching the last two letters to get acb. The next alphabetical order must start with b. The first possible listing starting with b must follow b with a, leaving c for last. The orders therefore bac. The only other order starting with b is bca. The next order must start with c, which will be followed by a to give us cab. The next order is obtained by switching the last two letters to get cba. This exhausts all possibilities for combinations of the three letters a, b and c. Our combinations are, in alphabetical order, abc, acb, bac, bca, cab, cba.......!!!!!!!!...................................
RESPONSE --> Ok, I got this one right, too.
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10:41:26 `q003. One collection consists of the letters a, c, d and f. Another collection consists of the letters a, b, d and g.
List the letters common to both collections. List the letters which appear in at least one of the collections. List the letters in the first half of the alphabet which do not appear in either of the collections.......!!!!!!!!...................................
RESPONSE --> Common letters: A and D
Letters that appear in at least one fo the collections: A, B,C,D,F, and G. Letters that do not appear: E,H,I,J,K,L, and M.................................................
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10:41:47 To letters a and d each appear in both collections. No other letter does.
The letters a, c, d, and f appear in the first collection, so they all in at least one of the collections. In addition to letters b and g appear in the second collection. Therefore letters a, b, c, d, f and g all appear in at least one of the collections. We consider the letters in the first half of the alphabet, in alphabetical order. a, b, c and d all appear in at least one of the collections, but the letter e does not. The letters f and g also appear in at least one of the collections, but none of the other letters of the alphabet do. The first half of the alphabet ends at m, so the list of letters in the first half of the alphabet which do not occur in at least one of the collections is e, h, i, j, k, l, m.......!!!!!!!!...................................
RESPONSE --> Ok
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10:43:36 `q004. Give the next element in each of the following patterns and explain how you obtained each:
2, 3, 5, 8, 12, ... 3, 6, 12, 24, ... 1, 3, 4, 7, 11, 18, .........!!!!!!!!...................................
RESPONSE --> 1) 20-because 8 + 12 =20
2) 48-becuase 24 * 2 = 48 3) 29-because 11+18=29.................................................
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10:45:46 The pattern of the sequence 2, 3, 5, 8, 12, ... can be seen by subtracting each number from its successor. 3-2 = 1, 5-3 = 2, 8-5 = 3, 12-8 = 4. The sequence of differences is therefore 1, 2, 3, 4, ... . The next difference will be 5, indicating that the next number must be 12 + 5 = 17.
The pattern of the sequence 3, 6, 12, 24, ... can be discovered by dividing each number into its successor. We obtain 6/3 = 2, 12/6 = 2, 24/12 = 2. This shows us that we are doubling each number to get the next. It follows that the next number in the sequence will be the double of 24, or 48. The pattern of the sequence 1, 3, 4, 7, 11, 18, ... is a little obvious. Starting with the third number in the sequence, each number is the sum of the two numbers proceeding. That is, 1 + 3 = 4, 3 + 4 = 7, 4 + 7 = 11, and 7 + 11 = 18. It follows that the next member should be 11 + 18 = 29.......!!!!!!!!...................................
RESPONSE --> I got the first one wrong because I saw the sequence to be (ex. 2+3=5, 3+5=8, ) Oh, Ok, I made an addition mistake. Thats where my problem is.
I got the next 2 correct..................................................
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10:49:36 `q005. The number 18 can be 'broken down' into the product 9 * 2, which can then be broken down into the product 3 * 3 * 2, which cannot be broken down any further . Alternatively 18 could be broken down into 6 * 3, which can then be broken down into 2 * 3 * 3.
Show how the numbers 28 and 34 can be broken down until they can't be broken down any further. Show that there at least two different ways to break down 28, but that when the breakdown is complete both ways end up giving you the same numbers.......!!!!!!!!...................................
RESPONSE --> 28 2 * 14 2 * 2 * 7
28 4 * 7 2 * 2 * 7 34 2 * 17.................................................
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10:50:00 A good system is to begin by attempting to divide the smallest possible number into the given number. In the case of 34 we see that the number can be divided by 2 give 34 = 2 * 17. It is clear that the factor 2 cannot be further broken down, and is easy to see that 17 cannot be further broken down. So the complete breakdown of 34 is 2 * 17.
To breakdown 28 we can again divide by 2 to get 28 = 2 * 14. The number 2 cannot be further broken down, but 14 can be divided by 2 to give 14 = 2 * 7, which cannot be further broken down. Thus we have 28 = 2 * 2 * 7. The number 28 could also the broken down initially into 4 * 7. The 4 can be further broken down into 2 * 2, so again we get 28 = 2 * 2 * 7. It turns out that the breakdown of a given number always ends up with exactly same numbers, no matter what the initial breakdown.......!!!!!!!!...................................
RESPONSE --> Ok, I got this one correct.
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10:51:13 `q006. Give the average of the numbers in the following list: 3, 4, 6, 6, 7, 7, 9. By how much does each number differ from the average?
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RESPONSE --> The average is 6. The difference from: 3 is 3 7 is 3 9 is 6
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10:51:37 To average least 7 numbers we add them in divide by 7. We get a total of 3 + 4 + 6 + 6 + 7 + 7 + 9 = 42, which we then divide by 7 to get the average 42 / 7 = 6.
We see that 3 differs from the average of 6 by 3, 4 differs from the average of 6 by 2, 6 differs from the average of 6 by 0, 7 differs from the average of 6 by 1, and 9 differs from the average of 6 by 3. A common error is to write the entire sequence of calculations on one line, as 3 + 4 + 6 + 6 + 7 + 7 + 9 = 42 / 7 = 6. This is a really terrible habit. The = sign indicates equality, and if one thing is equal to another, and this other today third thing, then the first thing must be equal to the third thing. This would mean that 3 + 4 + 6 + 6 + 7 + 7 + 9 would have to be equal to 6. This is clearly not the case. It is a serious error to use the = sign for anything but equality, and it should certainly not be used to indicate a sequence of calculations.......!!!!!!!!...................................
RESPONSE --> Ok
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10:53:43 `q007. Which of the following list of numbers is more spread out, 7, 8, 10, 10, 11, 13 or 894, 897, 902, 908, 910, 912? On what basis did you justify your answer?
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RESPONSE --> The second set is more spread out because there is a difference of 18 between the first number and the last number. The first set is only spread between 6 numbers.
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10:53:57 The first set of numbers ranges from 7 to 13, a difference of only 6. The second set ranges from 894 to 912, a difference of 18. So it appears pretty clear that the second set has more variation the first.
We might also look at the spacing between numbers, which in the first set is 1, 2, 0, 1, 2 and in the second set is 3, 5, 6, 2, 2. The spacing in the second set is clearly greater than the spacing in the first. There are other more sophisticated measures of the spread of a distribution of numbers, which you may encounter in your course.......!!!!!!!!...................................
RESPONSE --> Ok
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10:56:05 `q008. 12 is 9 more than 3 and also 4 times 3. We therefore say that 12 differs from 3 by 9, and that the ratio of 12 to 3 is 4.
What is the ratio of 36 to 4 and by how much does 36 differ from 4? If 288 is in the same ratio to a certain number as 36 is to 4, what is that number?......!!!!!!!!...................................
RESPONSE --> The ratio of 36 is 4 by 9 and the difference is 32.
The number is 32..................................................
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10:56:13 Just as the ratio of 12 to 3 is 12 / 3 = 4, the ratio of 36 to 4 is 36 / 4 = 9. 36 differs from 4 by 36 - 4 = 32.
Since the ratio of 36 to 4 is 9, the number 288 will be in the same ratio to a number which is 1/9 as great, or 288 / 9 = 32. Putting this another way, the question asks for a 'certain number', and 288 is in the same ratio to that number as 36 to 4. 36 is 9 times as great as 4, so 288 is 9 times as great as the desired number. The desired number is therefore 288/9 = 32.......!!!!!!!!...................................
RESPONSE --> Ok
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10:59:39 `q009. A triangle has sides 3, 4 and 5. Another triangle has the identical shape of the first but is larger. Its shorter sides are 12 and 16. What is the length of its longest side?
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RESPONSE --> I think the lenght is 20.
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11:00:38 ** You need to first see that that each side of the larger triangle is 4 times the length of the corresponding side of the smaller. This can be seen in many ways, one of the most reliable is to check out the short-side ratios, which are 12/3 = 4 and 16/4 = 4. Since we have a 4-to-1 ratio for each set of corresponding sides, the side of the larger triangle that corresponds to the side of length 5 is 4 * 5 = 20. **
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RESPONSE --> OK, I actually worked this one out in my head but I do understand the solution you have given here.
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Nӈх̀xP Student Name: assignment #001 004. Liberal Arts Mathematics
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22:37:44 Note that there are 4 questions in this assignment.
`q001. Let A stand for the collection of all whole numbers which have at least one even digit (e.g., 237, 864, 6, 3972 are in the collection, while 397, 135, 1, 9937 are not). Let A ' stand for the collection of all whole numbers which are not in the collection A. Let B stand for the collection { 3, 8, 35, 89, 104, 357, 4321 }. What numbers do B and A have in common? What numbers do B and A' have in common?......!!!!!!!!...................................
RESPONSE --> The numbers that B and A have in common are: 8, 89, 104, and 4321 because the problem states that any number in A has to be a whole numbe with at least one even digit and the numbers I listed above may could also be found in A.
The numbers B and A' have in common are 3, 35, and 357 because the numbers in A' are any numbers that are not found in A (which must contain at least one even digit)..................................................
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22:38:24 Of the numbers in B, 8, 89, 104, 4321 each have at least one even digit and so are common to both sets. 3 is odd, both of the digits in the number 35 are odd, as are all three digits in the number 357. Both of these numbers are therefore in A ' .
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RESPONSE --> OK
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22:45:27 `q002. I have in a room 8 people with dark hair brown, 2 people with bright red hair, and 9 people with light brown or blonde hair. Nobody has more than one hair color. Is it possible that there are exactly 17 people in the room?
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RESPONSE --> I don't think that it is possible because 8 people with dark brown hair, 2 people with bright red hair, and 9 people with light brown or blond hair equal 19 people.
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22:45:46 If we assume that dark brown, light brown or blonde, and bright red hair are mutually exclusive (i.e., someone can't be both one category and another, much less all three), then we have at least 8 + 2 + 9 = 19 people in the room, and it is not possible that we have exactly 17.
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RESPONSE --> OK
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22:49:53 `q003. I have in a room 6 people with dark hair and 10 people with blue eyes. There are only 14 people in the room. But 10 + 6 = 16, which is more than 14. How can this be?
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RESPONSE --> There are 4 people who have both dark hair and blue eyes, 6 people who have only blue eyes, 2 people who have only dark hair, and 2 people who have neither. This give you a total of 14 people.
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22:57:35 The key here is that there is nothing mutully exclusive about these categories-a person can have blue eyes as well as dark hair. So if there are 2 people in the room who have dark hair and blue eyes, which is certainly possible, then when we add 10 + 6 = 16 those two people would be counted twice, once among the 6 blue-eyed people and once among the 10 dark-haired people. So the 16 we get would be 2 too high. To get the correct number we would have to subtract the 2 people who were counted twice to get 16 - 2 = 14 people.
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RESPONSE --> Ok, I came up with a different solution which does work for this problem but, I do see how you got the solution you have given. I actually drew a diagram like this: B=blue eyes H=dark hair O=different color eyes
BH BH BH BH B B B B B B OH OH O O I didn't recall the problem actually stating that each person has at least dark hair or blue eyes though. So I think that both solutions could be correct..................................................
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23:02:28 `q004. In a set of 100 child's blocks 60 blocks are cubical and 40 blocks are cylindrical. 30 of the blocks are red and 20 of the red blocks are cubical. How many of the cylindrical blocks are red?
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RESPONSE --> I think that 10 of the cylindrical blocks are red becuse if there are 100 blocks, (60 cubical and 40 cylindrical), 30 red blocks in which 20 of these are cubical, that leaves 10 to be cylindrical.
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23:02:40 Of the 30 red blocks 20 are cubical, so the rest must be cylindrical. This leaves 10 red cylindrical blocks.
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RESPONSE --> OK
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}Q̿{ Student Name: assignment #002
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23:11:45 Note that there are 2 questions in this assignment.
`q001. We can represent the collection consisting of the letters a, b, c, d, e, f by a circle in which we write these letters. If we have another collection consisting of the letters a, c, f, g, k, we could represent it also by a circle containing these letters. If both collections are represented in the same diagram, then since the two collections have certain elements in common the two circles should overlap. Sketch a diagram with two overlapping circles. The two circles will create four regions (click below on 'Next Picture'). The first region is the region where the circles overlap. The second region is the one outside of both circles. The third region is the part of the first circle that doesn't include the overlap. The fourth region is the part of the second circle that doesn't include the overlap. Number these regions with the Roman numerals I (the overlap), II (first circle outside overlap), III (second circle outside overlap) and IV (outside both circles). Let the first circle contain the letters in the first collection and let the second circle contain the letters in the second collection, with the letters common to both circles represented in the overlapping region. Which letters, if any, go in region I, which in region II, which in region III and which in region IV?......!!!!!!!!...................................
RESPONSE --> Region I: A,C, and F Region II: B,D, and E Region III: G and K Region IV: included all other letters not listed (example: H,I,J,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y, and Z)
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23:13:44 The letters a, c and f go in the overlapping region, which we called Region I. The remaining letters in the first collection are b, d, and e, and they go in the part of the first circle that does not include the overlapping region, which we called Region II. The letters g and k go in the part of the second circle that does not include the overlapping region (Region III). There are no letters in Region IV.
Click below on 'Next Picture' for a picture.......!!!!!!!!...................................
RESPONSE --> Ok, my problem is that I assumed that all the other letters not listed went into Region IV. So I guess that I should have only placed the listed letters into regions.
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23:24:01 `q002. Suppose that we have a total of 35 people in a room. Of these, 20 have dark hair and 15 have bright eyes. There are 8 people with dark hair and bright eyes.
Draw two circles, one representing the dark-haired people and the other representing the bright-eyed people. Represent the dark-haired people without bright eyes by writing this number in the part of the first circle that doesn't include the overlap (region II). Represent the number of bright-eyed people without dark hair by writing this number in the part of the second circle that doesn't include the overlap (region III). Write the appropriate number in the overlap (region I). How many people are included in the first circle, and how many in the second? How many people are included in both circles? How many of the 35 people are not included in either circle?......!!!!!!!!...................................
RESPONSE --> There should be 12 people in the first circle, and 7 people in the second cirlce.
There should be 8 people included in both circles. There should be 8 people that are not included in either circle. I came to this conclusion based on the solution of a previously worked problem. I took into consideration this time that some people may have been counted twice (those with dark hair and bright eyes)..................................................
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23:24:43 Of the 20 dark-haired people in the preceding example, 8 also have bright eyes. This leaves 12 dark-haired people for that part of the circle that doesn't include the overlap (region I).
The 8 having both dark hair and bright eyes will occupy the overlap (region I). Of the 15 people with bright eyes, 8 also have dark hair so the other 7 do not have dark hair, and this number will be represented by the part of the second circle that doesn't include the overlap (region III). We have accounted for 12 + 8 + 7 = 27 people. This leaves 35-27 = 8 people who are not included in either of the circles. The number 8 can be written outside the two circles (region IV) to indicate the 8 people who have neither dark hair nor bright eyes (click below on 'Next Picture').......!!!!!!!!...................................
RESPONSE --> Ok, it looks like I got this one right.
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