# proof

1. ### Continuous Random Variable Proof

Question: If X is a random variable for which P(X \geq 0)=0 and |\mu_x|<\infty, show that P(X<\mu t) \geq 1 - \frac{1}{t} for every t \geq 1. My TA told me to use the following information: \mu=\int_{-\infty}^{\infty}(x(f(x))dx Since P(X \geq 0)=0: \mu=\int_{0}^{\infty}(x(f(x))dx...
2. ### Help with a hyperbolic trig proof

I need help with this proof. Homework was online and I've already gotten it wrong but I still want to figure it out before moving on. Any help would be appreciated! Here is the proof... arcsin(tanhx)=arctan(sinhx)
3. ### Using Congruent Triangles (Two-Column Proof)

In this exercise, there is one piece of unnecessary information. State what information you do not need for the proof. Then give a two-column proof that does not use that piece of information. Given: LM is congruent to LN; KM is congruent KN; KO bisects MKN; Prove: LO bisects MLN. This...
4. ### False proof by induction

There's this classic proof by induction that is FALSE: The claim is that cars are all the same colour. This is equivalent to the claim that any set of cars must contain cars of the same colour. The claim is trivially true for the base case of n = 1 cars since, after all, a car has only one...
5. ### Example as a proof

Hi, I hope someone can help. I'm trying to determine whether the following statements can be proven merely by an example? It doesn't matter whether the statements are actually true, let's just assume for this exercise that they are true. As follows: Statement 1: "Powers of primes increase...
6. ### More Inductive Proof Questions

I want to have a little 'le cry'. Earlier today, I made a super detailed and tidy post on my question, but for some reason it didn't post(I think it was because of my internet), and now I've got to type it all again, and with less care and attention to detail. :( Anyways, not so long ago, I had...
7. ### Confusion in Inductive Proofs

Hey guys, I've been lurking around this site for a while, but I decided to finally stick in a question. Well, two, but I'll post one first and see if I can figure out the next one with the advice or help I get. I will post the question and then my working out and then what I'm confused about...
8. ### Formal Language Theory Help Continued- Grammar

I need help with these grammar problems: 1- S → AB 2- A → f 3- A → g 4- B → y 5- B → z Remember that the names of the non-terminal elements (the letters S, A, and B) are chosen arbitrarily, although by convention the letter S is used for the initial state of the string. This grammar...
9. ### Trig Proof of Sin2x

Hi, I hope someone can help me with this proof that I have. I understand the steps that would lead Sin2x = (2Cotx)/(1+Cot^2x) to equal 2CosxSinx. However, how did someone realize that Sin2x = (2Cotx)/(1+Cot^2x) in the first place? Do I just assume that is a thing in mathematics? My math...
10. ### Proving trig relationship

Hi, I hope someone can help. I was hoping that someone can provide a proof for why 1 + tan^2θ does not equal sec^2θ. I know that I can have θ equal something, and then see if both sides end up being the same e.g. 1 + tan^2(pi/3) = sec^2(pi/3) .... 1 + (√3)^2 = (1/(1/2))^2 ... 4 = 4. While...
11. ### Prove that V is a subspace of Rn

Let V = {x E Rn | Ax= λx} where A is an n x n matrix and λ is a real scalar, together with the usual operations for vector addition and scalar multiplication from Rn. Prove that V is a subspace of Rn.
12. ### Exponential generating functions help understanding

Use exponential generating functions to prove that: P(n) = P(n-1) + n, with n >= 1 and P(0) = 1 will result in this equation: P(n) = (1/2)n2 + (1/2)n + 1 I know how to do this with generating functions, you just plug in n=1, n=2, n=3,.... P(1) will result in (1/2)(1)2 + (1/2)(1) +1 which is 2...
13. ### Proving Summation is equal to expression for all positive integers n

Prove that ∑ from k =1 to n of 1/(4k^2 - 1) = n/(2n+1) for all positive integers n. In words: Prove that the sum (from k =1 to n) of one over (4 times k squared minus one) is equal to n over two times n + 1 I am beyond beyond beyond lost.
14. ### Proving a sum is equal to an expression for all positive integers n

Prove that ∑ from k =1 to n of (-1)^k*k^2 = (-1)^n*((n(n+1)/2) for all positive integers n. In words: Prove that the sum (from k =1 to n) of negative one to the k times k squared is equal to negative one to the n, times (n times (n plus 1), divided by 2. I am beyond lost.
15. ### Proof that A^n < N!

For any positive integer A, there is an integer N, which of course can vary depending on A, so that for all n >= N, A^n < n!. How am I supposed to show this? Do I need to show A^n = Big-O of n! or use induction? I tried both and I can't seem to figure it out.
16. ### Showing/proving multiplication using letters/variables

Let a, b, c be integers with a ≠ 0. Assume a | (b+c) and a | (b-c). Show that 1. a | (2b) 2. a | (2c) 3. a^2 | (b^2 -c^2) In words: "Let a, b, c be integers with a ≠ 0. Assume that a divides (b+c), and that a divides (b-c). Show that: #1 a divides (2 times b) #2 a divides...
17. ### More Friendly "Optional" Proof Questions - Part 1

Let A, B, C⊆ U. Assume (A∩C) ⊆B. Show that (A - B)⊆ (A - C). So here is what I have so far: "Let L= A-B and R=A-C. For A-B⊆ A-C, then L⊆R. Show L⊆R. Assume XeL, and let XeA. Either XeA or X∉B." My question (among looking for a clear cut proof) is should I actually start by stating that because...
18. ### Proving the Cardinality of 2 finite sets

Hi all! This is my first post, and my partner and I have struggled to try to understand the following optional/suggested problem: Let A and B be finite sets. Prove that |A - B| = |A u B| - |B|. In words: "Prove that the absolute value (or cardinality) of A minus B is equal to the absolute value...
19. ### Proof of a Sequence and serie

Dear Ladies and Gentlements, My Prof. give me this mathematical problem and i have no clue to solve this problem. Maybe someone find a way. Task The series a0, a1, a2... is defined as this instruction: a0=1 and an = an-1 * ( 4 - (2/n)) for n>= 1 now i need to prove that for evey n>=1...
20. ### Is the folowing statement true?

Let V be a vector space and W be its subspace. Is it true that the union of subspace W and its orthogonal complement equals V? If either no or yes, could you help me with the proof? Thanks.