TT's Maths Thread

[TrueTears]

I found the error, it should be f: Pn -> {1,2,....,m}

it means that the functions we are counting have to satisfy the property that:

if you have two sets A and B,

and you apply f to the intersection of them,

then the result is the same as if you

applied f to both of them seperately,
then picked the smallest output.

Oh so the question should read:

Let be the set of subsets of . Let be the number of functions such that . Prove that .

Is that right?

[addikaye03]

1) When two resistors of resistances z1 and z2 are connected in series, their equivalent resistance z is given by z=z1+z2, and when they are connected in parallel their equivalent resistance z is given by 1/z=1/z1+1/z2.

In an electric circuit that contains two resistors whose resistances are z1=R1+iwL and z2=R2-i/(wC) respectively, find the value of w so that their equivalent resistance is purely real when their resistors are

i) Connected in Series
ii)Connected in Parallel

2)Decompose the integrand into partial fractions with complex linear denominators, hence, find

int. dx/(x^8-1)

* i got the answer with plenty of tedious algebra + noting (x^4-1)(x^4+1) Hence roots of unity of (x^8-1), but want a quicker way*

3) A car of mass M kg, width w metres and centre of mass h metres above the ground travels round a level curve of radius r metres with a constant speed v m/s such that driver's right-hand side is near centre of the curve.

a) By drawing the front view of the car and two normal reactions of the road's surface on the right and the left wheels (neglect length of car), show that the right and left normal reactions respectively are:

M/2rw(rgw-2hv^2) and M/2rw(rgw+2hv^2)

b) hence, show that the car overturns when v^2>= rgw/2h

Are u joking?
What part of TT'S Maths thread dont you understand

All 3 Questions can be found within 'extension exercises' in 'Terry Lee MATHEMATICS Extension 2 HSC', i don't see what the big deal is:

Q1. Complex Numbers (Included in HSC+VCE course)
Q2. Integration (Included in HSC+VCE course)
Q3. Mechanics (Included in HSC+VCE? course)

[TrueTears]

1) When two resistors of resistances z1 and z2 are connected in series, their equivalent resistance z is given by z=z1+z2, and when they are connected in parallel their equivalent resistance z is given by 1/z=1/z1+1/z2.

In an electric circuit that contains two resistors whose resistances are z1=R1+iwL and z2=R2-i/(wC) respectively, find the value of w so that their equivalent resistance is purely real when their resistors are

i) Connected in Series
ii)Connected in Parallel

2)Decompose the integrand into partial fractions with complex linear denominators, hence, find

int. dx/(x^8-1)

* i got the answer with plenty of tedious algebra + noting (x^4-1)(x^4+1) Hence roots of unity of (x^8-1), but want a quicker way*

3) A car of mass M kg, width w metres and centre of mass h metres above the ground travels round a level curve of radius r metres with a constant speed v m/s such that driver's right-hand side is near centre of the curve.

a) By drawing the front view of the car and two normal reactions of the road's surface on the right and the left wheels (neglect length of car), show that the right and left normal reactions respectively are:

M/2rw(rgw-2hv^2) and M/2rw(rgw+2hv^2)

b) hence, show that the car overturns when v^2>= rgw/2h

http://vcenotes.com/forum/index.php/topic,20009.0.html

I asked some more Q here, if people would like to attempt them, they're interesting imo

There is no big deal, I love to see different types of questions, but it's just that you have already posted the questions and you posted it again. There's no need to bump something in such a short period.

[zzdfa]

nonono, it should be

I found the error, it should be f: Pn -> {1,2,....,m}

it means that the functions we are counting have to satisfy the property that:

if you have two sets A and B,

and you apply f to the intersection of them,

then the result is the same as if you

applied f to both of them seperately,
then picked the smallest output.

Oh so the question should read:

Let be the set of subsets of . Let be the number of functions such that . Prove that .

Is that right?

or else the answer wouldn't depend on n.

[TrueTears]

nonono, it should be

I found the error, it should be f: Pn -> {1,2,....,m}

it means that the functions we are counting have to satisfy the property that:

if you have two sets A and B,

and you apply f to the intersection of them,

then the result is the same as if you

applied f to both of them seperately,
then picked the smallest output.

Oh so the question should read:

Let be the set of subsets of . Let be the number of functions such that . Prove that .

Is that right?

or else the answer wouldn't depend on n.

Oh, then what does mean?


I get the bit now Thanks

[zzdfa]

f: A -> B just means a function called f that takes something from the set A and outputs something from the set B.
In this case, the set A is P_n and the set B is {1,2,....m}.

so this particular function takes in sets like {1,3,5} and outputs a number between 1 and m.

[TrueTears]

f: A -> B just means a function called f that takes something from the set A and outputs something from the set B.
In this case the elements of P_n are sets as well so it can be a little confusing.
so this particular function takes in sets like {1,3,5} and outputs a number between 1 and m.

Oh I see, thanks for that.

lol I don't even know where to start with this question. Completely stomped =(

[zzdfa]

It looks like it can be done with induction.

[TrueTears]

So let's say and

Then



So what's set A and set B? And what's ?


So can set A and B be anything?

If so then let and

Then

Then what is ?

[zzdfa]

edited my earlier post to clarify a bit.

so lets say that for some function f:P_3 -> {1,2,3,4}, f(   {1,2}    )= 2 and f(   {2,3}   )=3 and f(   {2}   )=4. Then this function is NOT counted because it doesn't satisfy the condition.

We want to know how many functions from P_3 to {1,2,3,4} DO satisfy the condition.


As an example, here are some easier questions:

how many quadratic functions f:R->R? infinite
how many quadratic functions f:R->R with zeroes at x=3 and x=5?    infinite
how many quadratic functions f:R->R with zeroes at x=3 and x=5 and coefficient of x^2 is 1?  one  

how many functions are there f: {1,2,3} -> {1,2,3}? what about {1,2,....n} ->{1,2,.....,m}?

what about the number of functions from P_n to {1,2,...,m}?

...

...

what about the number of functions from P_n to {1,2,....,m} with the property that f(A int B) = min{fA,fB}?

[zzdfa]

A,B are arbitrary elements from P_n.

So let's say and

Then



So what's set A and set B? And what's ?


So can set A and B be anything?

If so then let and

Then

Then what is ?

pretend you had a function that sent {1,2,3,4,5} to 4 and sent {2,3,7,8} to 3. Then {2,3} has to be sent to 3 because min(3,4)=3

[TrueTears]

A,B are arbitrary elements from P_n.

So let's say and

Then



So what's set A and set B? And what's ?


So can set A and B be anything?

If so then let and

Then

Then what is ?

pretend you had a function that sent {1,2,3,4,5} to 4 and sent {2,3,7,8} to 3. Then {2,3} has to be sent to 3 because min(3,4)=3

Ahh thanks, I think I have an idea now.

[zzdfa]

this problem is driving me crazy, fuck

[TrueTears]

Let be the set of subsets of . Let be the number of functions such that . Prove that .

lol I finally get this question, thanks zzdfa and Ahmad for your help.

Consider a specific case when and when

can arranged sets that contain the same number of elements from the highest to the lowest:









Now our aim is to find all functions such that

Let and



Now assume

Which means that . Contradiction!

This means that

Or more specifically the first row of must be larger than the 2nd row

This follows on that

Following on with our specific case of and

If was mapped to



Thus for there is only function which satisfies this.

If was mapped to

Then . This means the 2nd row has choices.

What about the 3rd row?

Consider

Say

This means that the value of the 3rd row is determined as soon as the values of the 2nd row is determined.

So if was mapped to there exist different function .

So overall for there exist different function .

Now for a more general result:

Consider the set .

This set can be arranged in the following way:







.
.
.





Now consider all the different mappings of to be

Consider mapped to 1.

Then there's 1 function.

If was mapped to 2. Then there would be different functions.

.
.
.

If was mapped to . Then there would be different functions.

In total we have

[Ahmad]

You've been nerd sniped haha