Query 2

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course MTH 151

2/18/13 5:00

Question: `q2.2.24 U={a,b,...,g}, A={a,e}, B={a,b,e,f,g}, C={b,f,g}, D={d,e}

Is the statement 'C is not a proper subset of U' true or false and why?

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Your solution:

False

It is a proper subset because all elements are in both C and U

confidence rating #$&*: 3

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Given Solution:

`a** The statement is false.

C is a proper subset of U because all elements of C are in the universal set, and because there are elements of U that aren't in C. Note that it takes both of these conditions to make U a proper subset of C, since a proper subset cannot be identical to other set. **

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Self-critique (if necessary):

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Self-critique Rating: OK

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Question: `qQuery 2.2.30 phi s D

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Your solution:

The statement is true because the book says that it is not possible to find any elements of an empty set that are not also in D

confidence rating #$&*: 3

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Given Solution:

`a** 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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Self-critique (if necessary):

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Question: `q(2.2.31, previously 2.2.33)

Is the following statement true or false:

D is not a subset of B

Is the statement true or false and why?

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Your solution:

D = {d, e}

B = { a, b, e, f, g}

The statement would be true because set B does not have a d in it like set D does

confidence rating #$&*: 3

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Given Solution:

`a** 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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Self-critique (if necessary):

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Question: `q2.2.34 (previously 2.2.36) there are exactly 31 subsets of B

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Your solution:

False

2^n

B has 5 elements in it so that would make it have 32 subsets

confidence rating #$&*: 3

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Given Solution:

`a** 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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Self-critique (if necessary):

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Question: `qQuery 2.2.38

Is the statement true or false and why?

There are exactly 128 proper subsets of U

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Your solution:

False

2^7=128

For this to be a proper subset you would have to subtract 1 which would equal 127

confidence rating #$&*: 3

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Given Solution:

`a** 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.

So thre are not 128 proper subsets of this set.**

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Self-critique (if necessary):

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Self-critique Rating: OK

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Question: `qQuery 2.2.45 U={1,2,...,10}. What is the complement of {1, 2, 3, 4, 6, 8}?

(previously 2.2.48 complement of {2,5,7,9,10} )

What is the complement of the given set?

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Your solution:

{5,7,9, 10}

confidence rating #$&*: 3

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Given Solution:

`a** the complement is {1,2,3,4,6,8}, the set of all elements in U that aren't in the given set.

The elements 5, 7, 9and 10 are not in the given set but are in U, so the complement is the set

{5, 7, 9, 10}**

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Question: `qquery 2.2.59 in how many ways can 3 of the five people A, B, C, D, E gather in a suite?

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Your solution:

{a ,b ,c), {a, c, d}, {a, c, e},{a, b, d}, {a, b, e} , {a, d, e}, {b, c, e}, {b, d, e}, {b, c, d}, {c, d, e}

confidence rating #$&*: 3

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Given Solution:

`a** 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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Question: `q (previously 2.1.74) (formerly 2.1.72) This was not assigned, but you should be able to answer based on your work on similar problems: It is or is it not true that 2 is not not subset of {7,6,5,4}?

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Your solution:

2 is not a subset of {7,6,5,4}

2 is not a set so how can 2 be a subset of it

confidence rating #$&*: 3

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Given Solution:

`a** 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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Self-critique (if necessary):

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Self-critique Rating:

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Question: `qquery 2.2.59 in how many ways can 3 of the five people A, B, C, D, E gather in a suite?

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Your solution:

{a ,b ,c), {a, c, d}, {a, c, e},{a, b, d}, {a, b, e} , {a, d, e}, {b, c, e}, {b, d, e}, {b, c, d}, {c, d, e}

confidence rating #$&*: 3

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Given Solution:

`a** 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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Your solution:

2 is not a subset of {7,6,5,4}

2 is not a set so how can 2 be a subset of it

confidence rating #$&*: 3

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Given Solution:

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.

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Self-critique (if necessary):

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Self-critique Rating:

#*&!

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Question: `qquery 2.2.59 in how many ways can 3 of the five people A, B, C, D, E gather in a suite?

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Your solution:

{a ,b ,c), {a, c, d}, {a, c, e},{a, b, d}, {a, b, e} , {a, d, e}, {b, c, e}, {b, d, e}, {b, c, d}, {c, d, e}

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

`a** 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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Your solution:

2 is not a subset of {7,6,5,4}

2 is not a set so how can 2 be a subset of it

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

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.

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Self-critique (if necessary):

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Query 2

&#Very good work. Let me know if you have questions. &#