## Challenge II

Posted Aug5-13 at 06:03 PM by micromass

The newest challenge was the following:

Quote:
Prove that ##\mathbb{N}## contains uncountably many subsets ##(N_\alpha)_{\alpha \in \mathbb{R}}## such that ##N_\alpha\cap N_\beta## is finite if ##\alpha\neq \beta##.
This was solved by HS-Scientist. Here's his solution:

Quote:
For any $a \in [0.1,1)$, define $S_a=(\lfloor 10a \rfloor,\lfloor 100a \rfloor...,\lfloor 10^na \rfloor...)$. For example, $S_\frac{\pi}{10}=(3,31,314,...) ... Posted in Uncategorized Views 469 Comments 0 ## Challenge Problem Posted Aug3-13 at 04:53 PM by micromass Updated Aug7-13 at 01:45 AM by micromass I have recently posted a challenge in my signature. The challenge read as follows: Quote: Let ##f:[0,+\infty )\rightarrow [0,+\infty )## be a twice differentiable function such that ##f(0) = 0##. Prove that if ##f^{\prime\prime}(x)\leq 0## for all ##x##, then ##f(x+y)\leq f(x) + f(y)## for all ##x## and ##y##. The first answer I got was from Millenial. He gave the following correct solution: Quote: Let [itex]g(x) = f'(x)$. Then, the question
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Posted Jul8-13 at 07:28 AM by micromass
Updated Jul8-13 at 03:37 PM by micromass

If you're not careful with indefinite integration, then paradoxes soon arise. For example, consider the following:

$$\int 0 dx = 0 \int 1dx = 0(x + C) = 0$$

But we also know that the integral of ##0## is supposed to be ##\int 0dx = C##. What's going on?

As another example, consider ##\int \frac{1}{x}dx##. This is an easy to solve integral, but let's apply integration by parts on it, we get

[tex]\int \frac{1}{x} dx = \frac{1}{x} x...
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## Are the rules at this forum too strict??

Posted Feb7-12 at 12:24 AM by micromass

I have not been a mentor for very long, but I can see pretty clearly that we get quite a lot of complaints of people who think the PF-rules are too strict or that we should be more lax in enforcing the rules. It may not surprise any of you that I disagree with this. But let me try to explain my motives behind my disagreement.

First of all, there is only one person in this entire forum who decides what the rules are. That person is Greg. It is his forum. He is the one who spends a...
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