Assignments 12

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15:33:14 Query 2.1.12 counting #'s 4 to 14 List the elements of the set.

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RESPONSE --> {5,6,7,8,9,10,11,12,13}

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15:39:29 query 2.1.24 set builder for set of presidents between LBJ and Clinton

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RESPONSE --> Okay {Nixon, Ford, Carter, Reagan, Bush}

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15:40:46 ** 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} **

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RESPONSE --> That would have been easier, I was thinking of set.

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15:41:05 2.1.40 finite or infinite: set of rat #'s 0 to 1

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RESPONSE --> finite

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15:42:37 ** 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. **

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RESPONSE --> okay

Your response did not agree with the given solution in all details, and you should therefore have addressed the discrepancy with a full self-critique, detailing the discrepancy and demonstrating exactly what you do and do not understand about the given solution, and if necessary asking specific questions.

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15:47:23 2.1.48 n(A), A={x|x is a U.S. senator} What is n(A) and why?

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RESPONSE --> I do not understand this question.

You should be responding with the solution you obtained when you originally did these problems. The brief notations here are not meant to be understood except in the context of the book's statement of the problem.

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15:48:09 ** 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 --> alright. I thought it would be that simple but I was trying to make it too complicated.

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15:49:00 query 2.1.54 {x|x is neagtive number}

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RESPONSE --> well-defined

You need to explain your reasoning on all questions, but especially on questions like this one where there are only a small number of possible answers.

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15:49:31 ** 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. **

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RESPONSE --> okay

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15:51:26 2.1.72 true or false 2 not subset of {7,6,5,4}

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RESPONSE --> True

Again you need to explain your reasoning.

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15:51:58 ** 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. **

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RESPONSE --> okay

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15:52:25 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?

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RESPONSE --> False

This answer also needs to be completely justified.

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15:52:35 ** Specifically it's false because the element 12 is in C but not in B. **

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RESPONSE --> Okay

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15:53:56 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?

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RESPONSE --> True

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15:54:03 ** 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 --> okay

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15:54:16 Query 2.2.30 phi s D

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RESPONSE --> false

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15:55:51 ** 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. **

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RESPONSE --> okay

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15:57:57 2.2.33 D not s B Is the statement true or false and why?

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RESPONSE --> True, because every element of D is not listed in B.

Good. This is a good example of what I mean by justifying your answer.

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15:58:55 ** 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. **

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RESPONSE --> okay

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15:59:28 2.2.36 there are exactly 31 subsets of B

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RESPONSE --> False

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15:59:43 ** 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. **

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RESPONSE --> Good

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16:01:05 Query 2.2.40 there are exactly 127 proper subsets of U Is the statement true or false and why?

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RESPONSE --> True because There would be 128 subsets and you subtract one and that leaves 127 proper subsets.

Again, good justification.

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16:01:21 ** 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 --> Good

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16:03:30 Query 2.2.48 U={1,2,...,10}, complement of {2,5,7,9,10} What is the complement of the given set?

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RESPONSE --> {1,3,4,6,8}

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16:03:43 ** 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 --> okay

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16:06:08 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 --> 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)

Once more, very good.

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16:06:20 ** 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}**

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RESPONSE --> okay

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Good work overall. However see my notes about documentation (missing on some problems, very nicely done on others) and self-critique and let me know if you have questions.