-1
votes
1answer
51 views

Mean and Standard Deviation self thought problem

I am 13 years old trying to teach myself about standard deviation and was wondering how this problem would look like. I know I am young to be learning this but I was reading about this and got ...
3
votes
4answers
80 views

How to easily prove $x+\frac{1}{x} \ge 2 \quad ∀x\in ℝ^+$ [duplicate]

When I tried to solve some certain math problem (an inequation) for pivate exercise purposes, I had to prove that $x+\frac{1}{x} \ge 2 \quad ∀x\in ℝ^+$, I solved it with tools from differential ...
6
votes
3answers
106 views

What should I do if I don't know where to start?

Sometimes getting started on a problem seems to be the hardest part. Once you find something to sink your teeth into, everything goes smoothly. What are some good things to try when you're staring at ...
1
vote
1answer
54 views

A confusion(possible book mistake) about one of the proofs in Spivak's Calculus?

In Chapter 5 - Function Limits, there is a proof that: if $|x - x_0| < 1; |x - x_0| < \frac{\epsilon}{2(|y_0| + 1)}; |y - y_0| < \frac{\epsilon}{2(|x_0| + 1)}$ then $|xy = x_0y_0| < ...
2
votes
2answers
496 views

Proving Holder's inequality using Jensen's inequality

Let $p$ and $q$ be positive reals such that $\frac{1}{p}+\frac{1}{q} = 1$, so that $p,q$ in $(1,\infty)$. For $\vec a$ and $\vec b \in \mathbb{R}^2$ prove that $|\vec a \cdot \vec b | \leq ||\vec ...
4
votes
0answers
148 views

Puzzle - zero knowledge proof

I am solving the following problem : I have edge-matching puzzles, where all pieces are squares and the grid has $n$*$n$ format. There is no global image to guide a puzzle solver. Despite the puzzles ...
7
votes
3answers
171 views

If $x_1, \ldots, x_6$ are positive real numbers that add up to $2$. Show that:

If $x_1,x_2,x_3,x_4,x_5$ and $x_6$ are positive real numbers that add up to $2$, then: $$2^{12} \leq \left(1+\dfrac{1}{x_1}\right) ...
1
vote
2answers
57 views

A non-equality and an inequality involving $y$ and $y_0$ from Spivak Calculus 4th ed.

It's Problem 22. from Chapter 1. I'm given: $y_0 \neq 0$ $|y - y_0| < \frac{|y_0|}{2}$ $|y - y_0| < \frac{\epsilon|y_0|^2}{2}$ and I must use them to prove that: $y \neq 0$ $|\frac{1}{y} - ...
0
votes
1answer
55 views

Pigeonhole Principle to solve question straightforward

A store wants to celebrate its anniversary and will give a $200 shopping certi cate to the first customer to enter the store whose birthday is the same as that of two other previously admitted ...
2
votes
2answers
139 views

how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$

I am given this equation: $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$ I want to prove it: what i did is I take any $a \in f^{-1}(B_1 \cap B_2)$, then there is $b \in (B_1 \cap B_2)$ so ...
0
votes
2answers
58 views

how to prove this: $f(A)=B$

I am given two sets: $A$ and $B$ and a function $f: A \rightarrow B$. I am asked to show and prove whether $f(A)=B$ is true or false. I am stuck not knowing how to do this. How can I do this?
0
votes
1answer
892 views

The cardinality of the power set with $N$ elements is equal to $2^N$ [duplicate]

Let $\mathcal{P}(X_N)$ be the power set of a set $X$ with $N$ elements. I am trying to prove by induction that its cardinality $\mid \mathcal{P}(X_N) \mid = 2^N$. Firstly, I think it helps to ...
0
votes
0answers
2k views

Prove that if the sum of two numbers is irrational then at least one of the numbers is irrational.

Question: Prove that if the sum of two numbers is irrational then at least one of the numbers is irrational. Is your proof direct, by contradiction, or by contrapositive? State the converse. Prove or ...
1
vote
1answer
299 views

Proving the transitivity of a relation

I want to prove that the relation $\sim$ on fractions given by $\frac{a}{b} \sim \frac{c}{d}$ if $ad = cb$, where $a, c \in \mathbb Z$ and $b, d \in \mathbb Z_{> 0}$, is transitive. (My last ...
1
vote
2answers
67 views

Exercise 1(d) from Courant

I'm having trouble understanding this "hint" in the back of (the first volume of) Courant's Differential and Integral Calculus text, which I'm just starting: One of the "challenging" Chapter 1 ...
0
votes
1answer
61 views

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if a>0 and b<0

Find the value of $\sqrt{(b-a-4)^2}- \sqrt{(a-b+1)^2}$ if $a>0$ and $b<0$. How do i find the value? This doesn't make any sense.
3
votes
2answers
129 views

choosing $5$ non consecutive books from a shelve of $12$

In how many ways can you pick five books from a shelve with twelve books, such that no two books you pick are consecutive? This is a problem that I have encountered in several different forms ...
4
votes
0answers
71 views

Arbitrary ratio sequences on a partition of $\mathbb{R}$ (Partition regularity of fixed ratio sequences)

Background: This question arose purely recreationally and doesn't really fit into any context that I know of. Let $A \sqcup B = \mathbb{R}$ be a partition of the ...
12
votes
4answers
1k views

Proving identities like $\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}$ combinatorially

I have to give a combinatorial proof of $$\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}.$$ I find it difficult to solve such problems. I'm not a brilliant person and never will be so I need to have ...
0
votes
1answer
213 views

Uniqueness of minimum and maximum of set

i am stuck in this simple but foggy problem. i need to prove or show that the min and sup are unique if they exist. $A$ is a nonempty set and $B$ is nonempty subset of $A$. i am trying to show that ...
1
vote
1answer
83 views

Squared Series Fourier [duplicate]

Possible Duplicate: Fourier 1st step? How to find fourier transform of a series of the such form: $$y_k=\left[f(x) \right]^{2},$$ but I am not sure of the step by step for going about this ...
4
votes
4answers
665 views

Serious applications of Colouring proofs

Are there any research-level applications of proofs by colouring? This is the kind of proof you use to show that you can't cover a mutilated chessboard with 31 dominoes. Afaik, this technique ...