Meital

Can someone help me with the following:

I am trying to prove that if [a,b] is a subset of a topological space X, where X is linear continuum, then everysubset of [a,b] that is bounded above has to have the least upper bound property. Can someone help me with that?

If X is a linear continuum space how do we show that all convex

subsets of X must inheret the linear contiuum axioms, that is, they

have betweenness and least upper bound property. Thanks.

If we have X with order topology, and we have [a,b] subset of X. And if we know that we have U open in X, can we conclude that [a,b] /\ U is open in [a,b]? If so please explain.

ok nevermind my request

OK..I am not sure if this is correct, but I think I can use the basis B ={ 1, x+1, x^2 + 2x}, or just use the basis in P_2 = { 1, x, x^2}. Then, I computed:

T(1) = 1 + 0 = 1

T(x) = x + 1

T(x^2) = x^2 + 2x

Then, (1)_B = vertical ( 1 0 0) then (x+1)_B = vertical ( 1 1 0) and (x^2 + 2x) = vertical ( 0 2 1 ), so the matrix representation is : with the following 3 columns ( 1 0 0, 1 1 0, 0 2 1), is that right?

I am not allowed to use graphs or anything, it is for Advanced Cal class..all we know is the series of the cosine and sine functions, then using that and convergence of series I guess, we need to show | cos x| < or = to 1. I know it would be so obvious to use graphs! But unfortunately I cannot use them here.

assuming that the only thing I know is the series of the function cosine' date=' that is, just using the following I need to be able to solve the problem:

 

cos x = 1 + Sum ( -1)^n (x^2n)/2n! the sum for n = 1 to infinity

I am also trying to prove the same results for

sin x = sum (-1)^n-1 x^(2n-1) / (2n - 1)!.[/quote']

I am trying to solve the following problem:

Show that | cos x | <= 1, that is, I am trying to show that the absolute value of cos x is less than or equal to 1 for all values of x, assuming that the only thing I know is the series of the function cosine, that is, just using the following I need to be able to solve the problem:

cos x = 1 + Sum ( -1)^n (x^2n)/2n! the sum for n = 1 to infinity

I am also trying to prove the same results for

sin x = sum (-1)^n-1 x^(2n-1) / (2n - 1)!

For the cos x, I broke it into 3 parts, one for x < 1, x = 1, x >1.

For x < or = 1, we have cos x = 1 + sum (-1)^n* x^2n/2n!, and we know this is the alternating series and it gives 1 + ( the value of the sum = -1 + e^x), but for x < or = 1, the e^x goes to 0, so we end up having |cos x| < or = to 1 for x < or = to 1. Is this right? Also, how do I show that for x > 1. Please help me. Thanks in advance.

Hello all..I am really having HARD TIME working the following problem..it took me forever and I am about to give up..I need help or even a hint on how to solve it..it is linear algebra/finding eigenvalues' problem.

Let V = R[X]_2 be the real polynomials of degree at most 2. Let T ( a linear transformation from V to V) be defined as (Tf)(x) = f(x) + f '(x). Find all the eigenvalues of T and their geometric and algebraic multiplicities.

by the way..for the intersection sign i used /\ and some other times \cap. Also for Y minus Z i used Y-Z or Y\Z.

I would really appreciate it if someone can check the proof to a claim i made in a bigger proof..here is the whole thing and below it is the part i want you to check:

Let X be a space, Y subset of X, and Z subset of Y. Give Y the subspace topology the following hold:

(1) Suppose Y is closed in X. Then Z closed in Y implies Z closed in X.

(2)Suppose Y is open in X. Then Z open in Y implies Z open in X.

Let us start with (2). Denote by T the topology on X. Then the topology

on Y is {U \cap Y | U in T}, where \cap = intersection. Thus if

Z is open in Y then Z = U \cap Y for some U which is open

in X. But now U and Y are both open in X, so their intersection

is open in X. This establishes (2).

(1) is similar; just note that if Z is closed in Y then Y-Z is

open in Y; i.e. Y-Z = U \cap Y for some U open in X. that this implies Z = (X-U) \cap Y. Thus Z is closed in X.

Now the proof to the claim, the one I need you to check is:

Z = (X\U) /\ Y

I showed 2 inclusions:

let x be in Z, since Z is subset of Y then x is in Y, since Y is a closed set in X then Y is a subset of (X\U)..so Z is a subset of (X\U) /\ Y

Now let x be in (X\U) /\ Y, then x is in X and x is not in U and x is in Y, then we ca say that x is in Y\ U and since Z is a closed subset in Y, then x is in Z too. So (X\U) /\ Y is a subset of Z, so one can conclude that Z = (X\U) /\ Y. Please check my proof and correct me if i am wrong. Thanks

I know what I am trying to prove is very simple, but I am stuck and I need help..

In vector spaces, let V be a vector space over F, c in F and v in V. I am trying to prove the following:

c0 = 0

My proof:

I know the main idea here is to use 0 = v + (-v), but until this point, we don't know that -v = -1.v, we don't know that if v /= 0 then there is an inverse such that v*V^-1 = 1, so let's see..by wa of contradiction, assume c0 /= 0 then

c(v +(-v)) /= 0 I am stuck here..any suggestions?

will someone please explain to me, for example in S_3, how do we find all the possible permutations, then from that how we find those permutations ( even ones) that are in A_3. I am looking at an example and it says that A_3 = {(1), (1,2,3), (1,3,2)}, but how can be (1,2,3), (1), and (1,3,2) be even permutations? Isn't the order of (1,2,3) = 3 ? I wrote down all the possible permutations of S_3, which are 6. Then I wrote them as cycles, so I got:

(1)(2)(3) , (1 2)(3), (1 3)(2), (2 3)(1), (1 2 3), (1 3 2). Can someone tell me exactly how I can find the order of a permutation, and how I can find the elements of A_3. I know that A_3 will have 3 elements, since its order is 3!/2 = 3.

If R is a ring with unit element 1 and f is a homomorphism of R into an integral domain R' such that I(f) /= R. Prove that f(1) is the unit element of R'.

for 1, a in R, 1.a = a.1 = a. Then f(1.a) = f(1)f(a) = f(a)f(1)= f(a) ( since R' is commutative) so we see that f(1) is the unit element of R'. Please check my proof and let me know of any mistakes I made. thanks in advance.

Hello everyone,

I am trying to find the form of the binomial theorem in general ring; that is, find an expression for ( a+b)^n, where n is a positive integer. I tried to solve it by induction, but usually we use induction when we know the answer and we try to show it is true. But here, I don't think here this will work. So I tried to write few terms, for n =1 a +b for n = 2 we have a^2 + 2ab + b^2 = a^2 + ab + ba + b^2, and for n = 3 it is a^3 + aab + aba + abb + baa+ bab + bba + b^3, so it is like the terms of (a+b)^n is consisted of all the possible arrangements between a and b in a sum, and every term of the sum has n entries either all a, all b, or a combination of a and b. I know we can do that since if a and b are elements of a general ring then we have addition is associative, so is multiplication, and both distributive laws hold. but the thing is I don't know how to write that in a fomal way or say, have it down as a formula or expression. Can someone help me put all that together? Thanks in advance.

Is F_p the set of integers that are modulo p, given that p is some prime number?

Hello all..I am taking abstract algebra class and I am practicing problems dealing with ring theory. One of the problems I have is not being able to construct examples or provide counter examples. For some reasons, most of the counter examples turn out to be either S_3 or group of quaternions. Are there famous groups that one needs to know in order to be able to construct counter examples to famous theorems in group and ring theory? Also, I wonder if someone knows an example of an integral domain which has an infinite number of elements, yet is of finite characteristic?

The procedure is as follows:

You roll the die the first time and you see a number, then now you are trying to get all the other numbers to appear at least once. So you roll again, if it is the same number you had the first time, you will roll again to get one of the 5 other numbers, otherwise, you will roll again to see the rest 4 numbers. So I don't think the way you solved it is correct. I know every side has a probability of 1/6 to appear, but this doesn't mean that every time you roll you will get a different number. I still need help

hello, I am trying to work the following problem, but i know i am doing something wrong since my answer doesn't make sense.

Suppose we roll a die repeatedly until we see each number at least once. Let R be the number of rolls required. fin ER ( the mean value of R).

The way I did it is:

Each number has a probability p of being shown, and q of not. p = 1/6, and q = 1-p = 5/6.

ER = p + pq + pq^2 + pq^3 + pq^4 + pq^5 for the first # that will show

+ pq + pq^2 + pq^3 + pq^4 + pq^5 for 2nd one

+ pq^2 + pq^3 + pq^4 + pq^5 for 3rd one

+ pq^3 + pq^4 + pq^5 for 4th one

+ pq^4 + pq^5 for 5th one

+ pq^5 for 6th one

but when finding the value i got an answer close to 2 which doesn't make sense, will you please tell me what i am doing wrong..this is urgent please let me know asap

I want you to provide me with some examples of how to write a permutation as a product of disjoint cycles.

Will you please provide another example and show me how to find the order of sigma? Thanks in advance.

For the first one I cannot use the integral test because we haven't proven it yet. I believe that I need to show that the sequence of partial sums is bounded to show that the series converges, I tried hard but I couldn't finish my proof.

I am studying permutations for the first time in my life. I tried to solve the problem that mathsfun's posted. I think order of sigma is 1, since the disjoint cycles of his example are ( 1 3 10)( 2 4 5 7)( 6 8 ) (9), order of each respec: 3, 4, 2, 1 and the LCM of them is 1.

Hey guys, can someone help me with the following proofs:

Prove that the sum (1/n^p) for n=1 to infinity converges for p>1.

Also, can you help me in proving the ratio test, that is suppose that x_n > 0 for all n in N (N = set of natural numbers) and suppose also that lim sup ( x_n+1/x_n) < 1, then prove that the series x_n for n = 1 to infinity converges. Then if lim inf ( x_n+1/x_n) > 1 then the same series x_n diverges.

I am trying to solve some problems on distribution functions and expected values, and I am stuck with 4 of these problems. I will appreciate any help or hints. thanks in advance!

Suppose X_1 and X_2 are independent and normal(0,1). Find the distribution function of Y = ( X_1^2 + X_2^2)^1/2. This is the Rayleigh distribution.

Suppose X1...Xn are independent and have density function f(x). Let Y =min { X1,...,Xn} and Z= max{X1,...,Xn}. Compute P( Y >= y, Z=<z) and differentiate to find joint density of Y and Z.

Suppose X has the standard normal distribution. Compute E|X|.

Suppose X has the standard normal distribution. Use integration by parts to conclude that EX^k = (k-1)EX^(k-2) and then conclude that for all integers n, EX^(2n-1)=0 and EX^2n = (2n-1)(2n-3)...3.1.