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        riddles >> general problem-solving / chatting / whatever >> Set notation help and other things
        
(Message started by: mikedagr8 on Apr 20th, 2008, 5:17am)
Title: Set notation help and other things
Post by mikedagr8 on Apr 20th, 2008, 5:17am
G'day guys, I'm doing my holiday homework and I don't understand this question as well as a few others. Could you please help me out?

The question states:

For A = {3,4,7,8} and B = {2,3,4,6,10} describe:

B/A

Does this just mean I divide the elements? What does this actually mean, and how do I do it.

Thanks again.
Title: Re: Set notation help and other things
Post by ThudanBlunder on Apr 20th, 2008, 5:40am
I believe it means the set of members of A that are not in B.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 20th, 2008, 5:42am
Ah, that would make sense. Thanks TB.
Title: Re: Set notation help and other things
Post by Obob on Apr 20th, 2008, 6:11am
That is really bad notation, though!  One would typically denote by A\B the set of elements of A that are not in B.  The notation B/A often has an entirely different meaning:  it is the set B, with some of its elements identified according to the set A.  In this particular case, it would be B/A = {2,6,10,{3,4].  This notion makes more sense when A is contained in B though.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 20th, 2008, 6:17am
Do I write B\A not as {2,6,10} or as {2,6,10{3,4]?

So, if another question has A = {5,6,7,8,9} and B={2,5,6,9,138}

How would I write A\B
{7,8{5,6,9]    or     {7,8}.

Thanks again for the help.
Title: Re: Set notation help and other things
Post by ThudanBlunder on Apr 20th, 2008, 8:33am
Oh, it was B/A, not A/B.    :-[

In which case, I would say B/A denotes the set of members of B that are not in A.
But looking it up, I see that the back slash (\) is used.

Mike, are you sure about the forward slash (/)?
Are there other examples using a back slash?
Title: Re: Set notation help and other things
Post by Obob on Apr 20th, 2008, 9:41am
If this is a homework assignment based off a textbook, the only sure fire thing is to look up what notation the textbook uses.  If it was given by a teacher, ask the teacher what the notation means.  I would be surprised if what they mean by B/A is what I said in the last post.  Also make sure that it is in fact a forward slash / and not a backslash \.  
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 20th, 2008, 4:54pm

on 04/20/08 at 08:33:31, ThudanBlunder wrote:
Mike, are you sure about the forward slash (/)?
Are there other examples using a back slash?

Yes (to both quesations), it should be a back slash. At 11:30pm I forget what I am doing. It is corrected now.

Obob, I can't ask them right now, as they are on holidays for another 8 days.
Title: Re: Set notation help and other things
Post by Obob on Apr 20th, 2008, 5:03pm
The notation A\B definitely means the elements of A that are not in B.  So in the original example, the answer is {7,8}.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 21st, 2008, 4:20am

on 04/20/08 at 17:03:08, Obob wrote:
The notation A\B definitely means the elements of A that are not in B.  So in the original example, the answer is {7,8}.

Thanks Obob.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 21st, 2008, 6:19am
Here is some more complex kind of stuff. Yes the pun is intended.

For z=12+16http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/imath.gif w=5+12http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/imath.giffind:

|z|
and
|z+w|

When I type |z| on the calculator I get an answer of 20 and |z+w| 32.75667871..... How does this occur and why? Once again we have not been taught absolute functions of imaginary numbers.
Title: Re: Set notation help and other things
Post by gotit on Apr 21st, 2008, 7:09am
If z=a+bi, then |z|=sqrt(a2+b2)

If z=a+bi and w=c+di, then z+w=(a+c)+(b+d)i.

Hope that helps.
Title: Re: Set notation help and other things
Post by Obob on Apr 21st, 2008, 9:54am
If you plot the complex number z in the complex plane, then |z| is the length of the line segment joining 0 to z.  So |z| really does generalize the notion of absolute value of a real number, since the length of the line segment from 0 to a real number a is just |a|.

An are there really no sections of the textbook to go with these problems?  If not, it sounds like a terrible teacher, assigning problems with untaught notation over holiday...
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 21st, 2008, 6:04pm

on 04/21/08 at 09:54:19, Obob wrote:
are there really no sections of the textbook to go with these problems?  If not, it sounds like a terrible teacher, assigning problems with untaught notation over holiday...

No, he is a wonderful teacher. He just has the highest expectations of our work, and quite often as a class he is disappointed. He wants us to work some things out by ourselves. The chapter in the text book for this does not cover modulus functions of imginary values sufficiently. It does so once when explaining how to multiply and divide 'cis' values, but that is it.

Sorry about that.
Thanks again.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 21st, 2008, 6:36pm
How do I change these equations to polar form. We have done this, but nothing as difficult as this. I have attempted these questions, but could not find an appropriate answer (I feel). Thanks again for the help guys.

For y2=(x3)/(a-x)

I ended up with the equation r=(asin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)/(cos3http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif+sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif) Is this correct and if not what is the correct answer?
Title: Re: Set notation help and other things
Post by ThudanBlunder on Apr 21st, 2008, 11:25pm
Hint: (y/x)2 = x/(a-x)
Substitute for y/x, rearrange, and eliminate x.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 21st, 2008, 11:46pm

on 04/21/08 at 23:25:02, ThudanBlunder wrote:
Hint: (y/x)2 = x/(a-x)
Substitute for y/x, rearrange, and eliminate x.

I'm sorry, but I don't understand.  :'( What do you mean substitue for y/w?
Title: Re: Set notation help and other things
Post by ThudanBlunder on Apr 21st, 2008, 11:52pm

on 04/21/08 at 23:46:26, mikedagr8 wrote:
I'm sorry, but I don't understand.  :'( What do you mean substitue for y/w?

y = rsin(theta)
x = rcos(theta)
So (y/x) = tan(theta)
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 21st, 2008, 11:54pm

on 04/21/08 at 23:52:36, ThudanBlunder wrote:
So (y/x) = tan(theta)

I forgot that.  :'( Thanks.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 22nd, 2008, 12:00am
I now arrive at
r = (atan2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)/(coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif+tan2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)

Still doesn't look correct though. :-/
Title: Re: Set notation help and other things
Post by ThudanBlunder on Apr 22nd, 2008, 3:41pm

on 04/22/08 at 00:00:34, mikedagr8 wrote:
I now arrive at
r = (atan2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)/(coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif+tan2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)

Still doesn't look correct though. :-/

What does 1 + tan2(theta) equal?
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 22nd, 2008, 7:51pm

on 04/22/08 at 15:41:11, ThudanBlunder wrote:
What does 1 + tan2(theta) equal?

The only reason I know this is because I'm looking this up in the book. I have not learnt this yet but it is equal to sec2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif. How does this help?
Title: Re: Set notation help and other things
Post by ThudanBlunder on Apr 22nd, 2008, 7:55pm

on 04/22/08 at 19:51:20, mikedagr8 wrote:
How does this help?

It helps if you consider your denominator.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 22nd, 2008, 8:24pm
OK. Does atan2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif+a/(coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif) look better?
Have I made any other errors?

Ah, could it be atan2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif/(coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gifsec2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)?
Title: Re: Set notation help and other things
Post by ThudanBlunder on Apr 22nd, 2008, 8:32pm

on 04/22/08 at 20:24:03, mikedagr8 wrote:
OK. Does atan2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif+a/(coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif) look better?

No, a(sechttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif - coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif) looks better.
But it seems you don't really know what sechttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif means.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 22nd, 2008, 8:35pm

on 04/22/08 at 20:32:59, ThudanBlunder wrote:
No, a(sechttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif - coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif) looks better.
But it seems you don't really know what sechttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif means.

I've been getting help for a reason. :P Thanks TB.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 22nd, 2008, 8:47pm
Now onto a different topic.
How do I expressth e following first-order difference in terms of n:
tn+1=3tn+2, t1=10.

I've tried to do this once again, but I'm not getting anywhere.

Could I get more help please?

Thanks once again for the help you have been giving me.
Title: Re: Set notation help and other things
Post by Aryabhatta on Apr 22nd, 2008, 10:54pm

on 04/22/08 at 20:47:46, mikedagr8 wrote:
Now onto a different topic.
How do I expressth e following first-order difference in terms of n:
tn+1=3tn+2, t1=10.

I've tried to do this once again, but I'm not getting anywhere.

Could I get more help please?

Thanks once again for the help you have been giving me.


Can you solve this one?

sn+1=3sn, s1 = 11

Try out a few terms of both sn and tn.

Do you notice a pattern?
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 22nd, 2008, 11:58pm
The pattern of tn seems to be that they are 3tn-1. Sn = Sn-1 +t n. This is assuming I've done this correctly.
Title: Re: Set notation help and other things
Post by towr on Apr 23rd, 2008, 12:23am
If you know how to find the closed expression for a second order recursion you can try this:

tn+1=3tn+2, t1=10.
=>
t2=32, and  tn+1-3 tn=2, so 2=t(n)-3tn-1
=>
tn+1=3tn+t(n)-3 tn-1
=>
tn+1=4 tn - 3 tn-1, with t1=10, t2=32

It may not seem like an improvement, but at least I know how to solve it.
The solution in this case will be of the form tn = A l1n + B l2n, with l1,l2 the roots of l2= 4 l - 3, and A,B can be found from the first two values (t1=10, t2=32)

There, eh, may be a simpler way. But I work with what I have.


[edit]I suppose I do also have Aryabhatta's clue to work with
tn   sn
10   11
98   99
296  297
890  891

Seems clearly related..
But I'm not sure how I'd find out sn without it being given (aside from using my above method which, naturally, gives the same answer)
[/edit]
Title: Re: Set notation help and other things
Post by Aryabhatta on Apr 23rd, 2008, 8:45am
Yes towr, there is a simpler way.

To avoid spoiling it for you, I will hide it:

[hide]
we have tn+1= 3tn + 2

If only it was in the form sn+1 = 3sn

Is there a way we can remove the constant term from the original equation?

We can! put sn +x = tn
and substitue in the first one.

We get
sn+1 + x = 3n + 3x + 2

i.e we need x = 3x + 2. i.e. x = -1.

So if sn = tn + 1
then we have

sn+1 = 3sn

which is easily solvable.
[/hide]
Title: Re: Set notation help and other things
Post by towr on Apr 23rd, 2008, 9:45am

on 04/23/08 at 08:45:03, Aryabhatta wrote:
Yes towr, there is a simpler way.

To avoid spoiling it for you, I will hide it:
I had already solved the general problem (tn+1=a tn+b, t1=c) with my earlier solution. In general, once you know it's inevitably of a certain form ([hide]K*an+M[/hide]), finding out the right constants is easy enough.
Still, it's probably a good trick to know.
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 23rd, 2008, 6:19pm
I'm lost in a maze of confused here. :-[

OK, new question as I'm not making any headway. Thanks guys for the help, I'm sorry I didn't use it, as I didn't understand all of it.

tn+1=-3tn+4 , t1=6

How do I find the S5? I tried using the formula (n/2)(2a+(n-1)d) but it didn't work out due to the +4 at the end, and I also tried using (n/2)(a+l) but I still couldn't figure it out. The only way I could find it was by working out t1-t5 and then adding. Surely there must be a formula to do this, or have I just not used the formulae correctly?

Thanks again.
Title: Re: Set notation help and other things
Post by towr on Apr 23rd, 2008, 11:26pm

on 04/23/08 at 18:19:09, mikedagr8 wrote:
tn+1=-3tn+4 , t1=6

How do I find the S5?
What is S? Or do you mean t5?


In any case, given
tn+1=-3tn+4 , t1=6
We'll have tn = A (-3)n+B, for some constants A and B
t1=6, so A (-3)1+B=6
=> B = 6+3A
t2=-3*6+4=-14, so A (-3)2+B=-14
=> 9A + (6+3A) = -14
=> 12A = -20
=> A = -5/3
=> B = 6 + 3A = 6 - 5 = 1
tn = -5/3 (-3)n+1

check:
t1 = -5/3 (-3)1+1=6
tn+1 = -5/3 (-3)n+1+1 = -3 (-5/3 (-3)n) +1 =  -3 (-5/3 (-3)n) -3 +3 +1 =  -3 (-5/3 (-3)n +1) +4 = -3tn+4
Title: Re: Set notation help and other things
Post by mikedagr8 on Apr 24th, 2008, 4:14am
S5 is the sum of the first 5 terms I would have thought as it is a series? I'm just going by what the question says, there  may be an error in the question, I'm not sure.
Title: Re: Set notation help and other things
Post by ThudanBlunder on Apr 24th, 2008, 4:15pm
[quote author=mikedagr8 link=board=riddles_general;num=1208693870;start=25#32 date=04/23/08 at 18:19:09]I'm lost in a maze of confused here. :-[/quote]
Do too many kooks spoil the broth?   ::)


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