# All Questions

856 views

### The cartesian product $\mathbb{N} \times \mathbb{N}$ is countable

I'm examining a proof I have read that claims to show that the Cartesian product $\mathbb{N} \times \mathbb{N}$ is countable, and as part of this proof, I am looking to show that the given map is ...
581 views

### Number of permutations where n ≠ position n

I am trying to figure out how many permutations exist in a set where none of the numbers equal their own position in the set; for example, 3,1,5,2,4 is an acceptable permutation where 3,1,2,4,5 is not ...
367 views

### Solution of 3 equations in 3 unknowns [closed]

Find the value of $c$ which makes it possible to solve: $$u+v+2w=2,$$ $$2u+3v-w=5,$$ $$3u+4v+w=c$$
3k views

### How can you prove that the square root of two is irrational?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
2k views

### Will moving differentiation from inside, to outside an integral, change the result?

I'm interested in the potential of such a technique. I got the idea from Moron's answer to this question, which uses the technique of differentiation under the integral. Now, I'd like to consider ...
2k views

### The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$

I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$. ...
2k views

### Motivation behind standard deviation?

Let's take the numbers 0-10. Their mean is 5, and the individual deviations from 5 are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 And so the average (magnitude of) ...
1k views

### Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$?

Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$ I.e - does the second half of the harmonic series go to zero? I know that for a finite number of terms the limit of the sum is ...
2k views

### What is the term for a factorial type operation, but with summation instead of products?

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems) I'm aware of Sigma notation, but is there a function/name ...
5k views

### $\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis

Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals. I am aware of the calculation ...
8k views

### What is the most elegant proof of the Pythagorean theorem?

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
2k views

### Convergence a.e. and of norms implies that in Lebesgue space

Let $1\leq p < \infty$. Suppose that $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite), $f_k \to f$ almost everywhere, and $\|f_k\|_{L^p} \to \|f\|_{L^p}$. Why is it ...
917 views

### Classification of prime ideals of $\mathbb{Z}[X]$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? (1) ($0$) (2) ...
13k views

### Union of two vector subspaces not a subspace?

I'm having a difficult time understanding this statement. Can someone please explain with a concrete example?
3k views

### How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to ...
8k views

### Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b )$

I know it might be too easy for you guys here. I'm practicing some problems in the textbook, but this one really drove me crazy. From $\gcd( a, b ) = 1$, I have $ax + by = 1$, where should I go from ...