Building Sets

Date: 05/26/2002 at 10:06:50
From: crystalia
Subject: Elements of : {x:x is a multiple of.....}

This is the problem I don't get:

  list all elements of: {x:x is a multiple of 7 and 5 < x < 56}

I don't understand why 5 is in this set, because it says that x is a 
multiple of 7, and 5 is not a multiple of 7...

Does this mean that 5 is not included in the elements that I'll make?
I've tried to solve this problem, and I kind of did but I still don't 
get if I will have to include 5 in my set or not... Here is my 
current answer, and please tell me if it's right or wrong.

  answer: {7,14,21,28,35,42,56}

Thank you so much!

Date: 05/26/2002 at 11:12:07
From: Doctor Ian
Subject: Re: Elements of : {x:x is a multiple of.....}

Hi Crystalia,

The typical use of this form of notation is 

  {variable: generator and test}

That is, there are two parts to the specification.  The generator 
give you a way to generate candidates.  In this case, you can 
imagine generating a list of _all_ the multiples of 7:
 
   ..., -21, -14, -7, 0, 7, 14, 21, ...

and then subjecting each one to the test:

  5 < -21 < 56 ?      NO, so -21 is not in the set.

  5 < -14 < 56 ?      NO, so -14 is not in the set.

  5 <  -7 < 56 ?      NO, so  -7 is not in the set.

  5 <   0 < 56 ?      NO, so   0 is not in the set.

  5 <   7 < 56 ?      YES, so  7 it in the set.

  5 <  14 < 56 ?      YES, so 14 it in the set.

  5 <  21 < 56 ?      YES, so 21 is in the set.

Now, what happens when we get up around 56? 

  5 <  42 < 56 ?      YES, so 42 is in the set.

  5 <  49 < 56 ?      YES, so 49 is in the set.

  5 <  56 < 56 ?      NO, so  56 is not in the set.

  5 <  63 < 56 ?      NO, so  63 is not in the set.

Now, of course, by looking at the test, you can avoid generating 
candidates that will obviously be wrong.  There is no point 
thinking up multiples of 7 that are less than 5, or greater than 
56, so it would be natural to start at 7 and end at 56.  But you 
still have to subject _each_ element to the test!

It might help to see what's going on if you think in terms of 
something other than numbers.  Suppose we construct the following 
set:

  {x: x is a mature golden retriever
      and x is at least 8 inches long
      and x is no more than 22 feet long}

Does this mean that the set has to include a dog 8 inches long, 
or one 22 feet long?

If you see how the generator(s) and the test(s) fit together, you 
should be able to see why 

  {x: x is prime and 6 < x < 19}

refers to the set

  {7, 11, 13, 17}

and not to the set

  {6, 7, 11, 13, 17, 19}

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/26/2002 at 16:26:16
From: crystalia
Subject: Elements of : {x:x is a multiple of.....}

Thank you so much Dr.Math!!!!!

Now I get what it meant... thanks!!!

I love this Dr. Math website, it helps me understand math much much 
better.  It really makes me happy when you can't ask anybody else, 
like parents, but you guys! (You know how parents could be mad when 
you don't get something simple, hee hee.)

Thanks again, Dr.Math!!

crystalia