Query 22

course Mth 151

zЇp{ǃ׽assignment #001

Your work has been received. Please scroll through the document to see any inserted notes (inserted at the appropriate place in the document, in boldface) and a note at the end. The note at the end of the file will confirm that the file has been reviewed; be sure to read that note. If there is no note at the end, notify the instructor through the Submit Work form, and include the date of the posting to your access page.

yaHɮ

Liberal Arts Mathematics I

09-04-2006

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20:20:01

Query 2.1.12 counting #'s 4 to 14

List the elements of the set.

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RESPONSE -->

{5,6,7,8,...13}

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20:20:47

**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 didn't think that I would include 4 and 14 because the question asked for the set of numbers between 4 and 14.

The word 'between' is a bit ambiguous.

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20:21:23

query 2.1.24 set builder for set of presidents between LBJ and Clinton

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RESPONSE -->

{Nixon,Ford, Carter, Reagan, Bush}

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20:22:27

** 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 -->

Good grief! I listed as a set the list of presidents that actually served btn Johnson and Clinton. I need to read my instructions better I guess.

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20:23:22

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

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RESPONSE -->

{xIx is a rational number between 0 and 1}

is finite.

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20:23:51

** 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 -->

ok

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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20:26:18

2.1.48 n(A), A={x|x is a U.S. senator}

What is n(A) and why?

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RESPONSE -->

n(A)=100 because the number of elements of A are U.S. Senators and there would be 2 from each of the 50 states or 100.

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20:26:30

** 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 -->

finally got something right!

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20:27:15

query 2.1.54 {x|x is neagtive number}

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RESPONSE -->

{xIx is a negative number} is well defined because it can be ANY negative number.

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20:27:29

** 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 -->

OK

The set is 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.

This wasn't part of your answer and should have been noted in your self-critique.

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20:28:15

2.1.72 true or false 2 not subset of {7,6,5,4}

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RESPONSE -->

True because for 2 to be a subset of the set it would have to be an element of the set.

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20:28:39

** 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 -->

ok

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20:29:21

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 -->

That is false because 12 is an element of A, but is not an element of B>

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20:29:42

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

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RESPONSE -->

I think I put the wrong set names in my response, but the principle is the same.

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g^y

assignment #002

yaHɮ

Liberal Arts Mathematics I

09-04-2006

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20:31:02

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 -->

The statement is true because {b,f,g} are in C and in U but there are elements in U that are not in C.

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20:31:10

** 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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20:33:05

Query 2.2.30 phi s D

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RESPONSE -->

False because phi has no elements in it and D={d,e}

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20:34:15

** 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 -->

OK

This also requires a self-critique.

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20:35:55

2.2.33 D not s B

Is the statement true or false and why?

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RESPONSE -->

True because the elements of D are in B, but the elements of B are not in D.

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20:36:13

** 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 -->

OK

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20:37:47

2.2.36 there are exactly 31 subsets of B

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RESPONSE -->

False

we have to use 2^n with the n being the number of elements in the set. There are 5 elements in set B, so 2^5 = 32

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20:38:02

** 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 -->

ok

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20:39:19

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

To find the number of proper subsets we use 2^n - 1

with n being the number of elements in the set.

U has a total of 7 elements in the set so,

2^7 - 1 = 128 - 1 = 127

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20:39:24

** 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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20:42:12

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 -->

This question should be number 50 I think

A^1 = {1,3,4,6,8}

the complement of a set contains the elements of set U that are not included in set A.

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20:42:22

** 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

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20:44:20

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...

{ABC}

{ABD}

{ABE}

{ACD}

{ACE}

{ADE}

{BCD}

{BCE}

{BDE}

{CDE}

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20:44:29

** 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 -->

OK

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"

You did well on these questions overall. I did give you a couple of notes in the early part of the assignment about self-critique, but by the end you were doing a very good job of applying definitions and providing well-expressed answers. Let me know if you have questions.