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#1 Re: Help Me ! » Is this function discrete or continuous? » 2011-11-08 02:22:23

sumpm1
Au101 wrote:

I would actually agree with bobbym here. In mathematics and science, discrete objects, such as the integers, do not vary smoothly, they have distinct values, little blocks, as it were. With the real numbers, which are continuous, you can take two numbers which are as close together as you like, for example 3.111 and 3.112 and still have many numbers between these two. With your brownie recipe, I think that you are making individual chunks and so I would argue that it is discrete, it can't vary arbitrarily.

But that's just my take on the issue smile

As far as what you guys are saying about ice cube, brick, or brownie "being useful" is not the story though.

My point is this. If you printer takes 1 minute to print 6 flyers... Well half of a flyer is not useful, but you MUST agree that at SOME POINT a flyer was halfway done printing, that data STILL EXISTED, whether you find it useful or not!

So I have raised the point to my colleagues: Either SOMEONE or SOMETHING in the problem must disctate the discrete nature of the input or the output.

In those cases stated "an ice cube, a brick, a printed flyer..." If we are creating them, then we must admit that at some point they were "half way created" and all of the continuous data exists. But if the problem says "you are buying PRE MADE bricks..." then SOMEONE (the store employees) won't let you you break a brick in half or buy half of a brick. The store won't let you buy half of a PRE MADE discrete unit of brownie.

So I argue that SOMEONE (the people or the situation), or something (nature dictates that humans and apples exist as discrete growing organisms) must dictate this discrete nature.

And as far as going so in depth to say, could we actually use "one molecule of milk?" in our example, this is beyond the discussion i think. But others have brought this up and said that any data that is being measured or recorded in reality is discrete because we can only measure to certain approximations of values of time, weight... and so on. But I only see that as being true when the data is actually being measured. If I put on a blindfold and pour some milk into a container, even though I haven't determined a measurement for what I poured, I can create infinite values of milk. But if you asked me to measure 3.14... cups of milk, no I couldn't... so a big difference on "measurement vs possible values."

#2 Re: Help Me ! » Is this function discrete or continuous? » 2011-11-07 14:57:23

sumpm1
bob bundy wrote:

hi sumpm1

It all depends on whether you can make, say, half a brownie.  If it is only possible to make whole brownies, then the function is discrete but (and I should confess, I am no cook!) I had a look at a recipe

http://allrecipes.co.uk/recipe/6023/qui … wnies.aspx

and the key phrase is:

3.Cool, and slice into equal square portions.

Clearly, that means you can divide up your mix into any shape and size.

Then you are in the realms of philosophy.

For if you take a brownie and cut it in two, have you got two half brownies or just two brownies (albeit smaller)?

So it begins to look like you can argue this either way.  I think your innocent question is going to run and run.  smile

EDIT:  If you go on subdividing a brownie into smaller and smaller pieces, eventually you will have to stop when you reach the atomic level.  That would make it discrete.

Oh darn!  That means this thread will also get side-tracked into whether it is possible to 'split the brownie atom'.

According to Dictionary.com

a small, chewy, cakelike cookie, usually made with chocolate and containing nuts.

Small seems to imply that a split brownie becomes two brownies => a discrete function.

If you teacher fails to grasp this argument, say you made some brownies to bring in and, to make sure there were enough to feed the whole school, you divided them up into tiny slices.  Hold up an empty hand and offer the teacher his / her share.

Bob

Haha, thanks for going so in depth. I must confess, I am a 9th grade teacher and this was a problem on one of my warm-ups. Some of my colleagues saw the problem and argued for discrete, I argued for continuous. Here is my argument:

If you were to use 2 cups per the instructions, you would have 12 brownies, and 3 to 18 and so on; likewise 1/2 a cup would seemingly result in 3 brownies per the instructions, and 1/4 cup would then seem to make 1.5 of the brownies that the instructions are describing. Since we are making these brownies, we may end up with 3.5 of these brownies described in the instructions.

I compare the situation to the following:

1.  The directions on an ice cube tray say that one cup of water will make 6 cubes. (After you use 1/6 cup, the first cube slot is full, and at some point the second slot is 1/3 full and so on.)

2. The directions on a bag of concrete say that one pound of mix will make 6 bricks. (After you fill the form for the first brick, you begin filling the second, and so on.)

3. Your printer takes 1 minute to print 6 copies of a flyer. (At some point the second flyer is 5/6 of the way printed, and so on...)

#3 Help Me ! » Is this function discrete or continuous? » 2011-11-04 14:56:55

sumpm1
Replies: 9

Hello, had a simple discrete or continuous function question posed in my 9th grade algebra class. Here is the question:

Q: A recipe for brownies reads that 1 cup of milk will make 6 brownies. Is this function continuous or discrete?

What say you?

#4 Help Me ! » Real Analysis Help - urgent » 2009-04-28 13:43:03

sumpm1
Replies: 1











#5 Re: Help Me ! » Real Analysis Help » 2009-04-28 11:52:00

sumpm1


#6 Help Me ! » Real Analysis Help » 2009-04-25 00:29:58

sumpm1
Replies: 8











#7 Help Me ! » Real Analysis Help » 2009-04-20 14:48:43

sumpm1
Replies: 3























#8 Help Me ! » Real Analysis Help » 2009-04-10 03:30:58

sumpm1
Replies: 4









#9 Re: Help Me ! » Real Analysis Help » 2009-04-05 09:36:19

sumpm1



#10 Help Me ! » Real Analysis Help » 2009-04-04 03:06:34

sumpm1
Replies: 3











#11 Re: Help Me ! » Real Analysis Help » 2009-03-07 03:15:35

sumpm1

Hey guys, thanks for the help.

@ksmathwiz: Are you sure that the identity for cosh(2x) is not cosh(2x) = cosh^2(x) - sinh^2(x) rather than cosh(2x) = cosh^2(x) + sinh^2(x)?

Thanks

#12 Help Me ! » Real Analysis Help » 2009-03-05 23:13:39

sumpm1
Replies: 8

Hey guys. We are working on functions: increasing, decreasing, monotone, bounded, minimums, and maximums.



















#13 Help Me ! » Real Analysis Help » 2009-03-02 10:28:29

sumpm1
Replies: 2











#14 Help Me ! » Real Analysis Help » 2009-02-27 09:51:36

sumpm1
Replies: 4






#15 Re: Help Me ! » Real Analysis Help » 2009-02-22 07:11:43

sumpm1

I know they look easy, but I was wondering what the beginning of one of the questions from #2 would look like, any one of them would do. Thank you.

#16 Re: Help Me ! » Real Analysis Help » 2009-02-21 11:24:42

sumpm1
mathsyperson wrote:

I want to point out that the Archimedes Principle actually says that given a real x, some natural n exists such that n > x.

The way you have it, choosing 0=a<b would make it false.




#17 Help Me ! » Real Analysis Help » 2009-02-20 04:53:24

sumpm1
Replies: 8

Hi I need some more help in Real Analysis. We are still covering infimum and supremum of a set, as well as the Completeness Axiom, Archimedean Property, and the following theorems.

Completeness Axiom: Each nonempty set of real numbers that is bounded above has a supremum.

Theorem: Between any two distinct real numbers there is a rational number and an irrational number.

Archimedean Property of the Real Numbers: If

and
are real numbers, then there exists a positive integer
such that
.



Theorem (Follows from Archimedean): The following statements are equivalent.
    1. If

and
are real numbers, then there exists a positive integer
such that
.
    2. The set of positive integers is not bounded above.
    3. For each real number
, there exists an integer
such that
.
    4. For each real number
, there exists a positive integer
such that
.






I have these two questions:

1) Prove that a nonempty finite set contains its infimum.

2) Prove each of the following results - give a direct proof of each one - without the use of the Completeness Axiom. These results could be called the Archimedean Property of the rational numbers.

    a) If

and
are positive rational numbers, then there exists a positive integer
such that
.

    b) For each positive integer

, there exists a rational number
such that
.

    c) For each rational number

, there exists an integer
such that
.

    d) For each positive rational number

, there is a positive integer
such that
.


Any help on #1 would be greatly appreciated.

Thanks

#18 Re: Help Me ! » Real Analysis Help » 2009-02-16 02:07:28

sumpm1

Yes, I do need to prove that -gamma is the greatest lower bound of S, but how do I go about this?

#19 Re: Help Me ! » Real Analysis Help » 2009-02-13 23:25:06

sumpm1

Hey Luis, for one, you can check on some of my other posts to see where another university is at during the semester. Are you guys on a similar level at this point in the semester? It would be cool to have someone else tho chat with about this stuff. Thanks for the help everyone.

#20 Help Me ! » Real Analysis Help » 2009-02-13 10:55:48

sumpm1
Replies: 7

Hi I need some more help in Real Analysis. We are covering infimum and supremum of a set, as well as the Completeness Axiom.

Completeness Axiom: Each nonempty set of real numbers that is bounded above has a supremum.

1. The Completeness Axiom only asserts something about sets that are bounded above. Use the Completeness Axiom to prove that every nonempty set of real numbers is bounded below has an infimum.

2. Let

be a nonempty set of real numbers that is bounded above and let
. Prove that for each
there exists a point
such that
.

Where "supS" is the supremum of the set S.

Thanks

#21 Help Me ! » Real Analysis Help » 2009-02-09 05:15:35

sumpm1
Replies: 3

Hello, we are covering absolute values, intervals and inequalities in my Real Analysis class. Specifically, we have recently covered the Geometric Sum, as well as the Arithmetic Mean/Geometric Mean Inequality that are stated below.

Geometric Sum: If

and
are real numbers, then

Arithmetic Mean/Geometric Mean Inequality: Let

be a positive integer. If
are nonnegative numbers, then
. Equality occurs if and only if
.

Problems:

1. Let

and
be fixed positive numbers. Find the value of
that minimizes the given expression and determine the minimum value of the expression.

a.

b.

c.

2. Let

be a positive integer and let
be positive numbers. The harmonic mean of these numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers. Prove that the harmonic mean of a set of positive numbers is less than or equal to the geometric mean. When does equality occur?

I am unclear of the method for finding the maximum or minimum of an expression. Ant help is appreciated.

Thanks

#22 Re: Help Me ! » Real Analysis Help » 2009-01-31 09:47:45

sumpm1

Ahhh... My fault, I was thinking that letting

, but you were simply restating the given in the problemdizzy!!!

Thanks again.

#23 Re: Help Me ! » Real Analysis Help » 2009-01-31 05:45:41

sumpm1

Thank you so much Jane.

In #2, I had to take

Is this all correct?

And in #3 you found that

I am having trouble arriving at this and am finding instead that

, but I am not confident in my understanding of how to manipulate inequalities in this way.

For an example of my understanding of manipulating inequalities: We let

. So I would follow that
and then that
, is this correct? Also, It may seem kind of DINGBAT of me... but I am getting hung up when moving
to the right side of the inequality and leaving the space empty:
.

Also this is troublesome to me and I need a better understanding of how to manipulate inequalities. Or perhaps it is the misuse of an EQUALITY and an INEQUALITY together as in the following manipulation(more DINGBAT):

Am I going about this all wrong?

Thanks

#24 Help Me ! » Real Analysis Help » 2009-01-28 19:55:24

sumpm1
Replies: 5

Hey guys, need some more help here. We have been covering absolute values and the Triangle inequality theorem in class.

Triangle Inequality Theorem:

Reverse Triangle Inequality:

1. Let

and
be two nonempty sets or real numbers, and let
be a real number. Suppose that there exist positive real numbers
and
such that
for all
and that
for all
. Use the Triangle Inequality Theorem to prove that
for all
and
.

2. Let

be a nonempty set of real numbers, and let
be a nonzero real number. Suppose that
for all
. Prove that
for all
.

3. Let

and
be real numbers. Suppose that
for all positive numbers
. Prove that
.



In number 1, I thought maybe I could add multiples of the inequalities

and
. But I am not sure how I should use the Triangle Inequality to get to
.

In number 2, It seems like maybe I should be using the arithmetic mean

somewhere. And this is in this section of the book, but we have not covered this yet, and I doubt the professor would assign a problem that needed this without presenting it in class. So the  key is probably again the Triangle Inequality or the Reverse Triangle Inequality.

And problem 3 I don't have a clue on.

I would appreciate some hints on this if anyone can provide some insight. I feel like this is less about UNDERSTANDING the concepts, and more about RECOGNIZING TRICKS!!! Perhaps they are one in the same, but I always have trouble finding the "trick."

Thanks

#25 Re: Help Me ! » Real Analysis Help » 2009-01-24 23:36:58

sumpm1
Ricky wrote:

The "no common divisor" method works for radical 2, but not much else.  Instead, follow the same proof for radical 2, stopping when you get integers on each side of the equation.  Now use the fundamental theorem of arithmetic to say that the two sides of the equation can never be equal.

Hey Ricky. I was able to use the same proof for proving

is irrational in #1. First I thought it would be showing that both sides were even, but instead was able to show that both sides must be multiples of 6 to get the contradiction. I will look at the fundamental theorem of arithmetic.

Thanks guys

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