0
votes
1answer
24 views

Prove that function f has a local minima and maxima

$f:R->R, f(x) = (x^2+mx)e^-x$ Show that, for every m in R, the function f has a local ...
0
votes
2answers
33 views

Having trouble showing the cardinality of two infinite sets is the same

We just learned about Aleph-naught today and I read about it on wikipedia but I do not know how to go about solving this problem in my homework: Prove that N(natural numbers) has the same ...
0
votes
2answers
69 views

How to show by the Root Test that $\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$ converges or diverges

How do I show by the Root Test that $$\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$$ converges or diverges? This is what I have done so far. Since we take $\sum\limits_{i=1}^\infty \sqrt[n]{|a_n|}$, we ...
1
vote
3answers
162 views

Why are irrational numbers uncountable and rationals contable?

Question 1: Why are irrational numbers uncountable and rationals contable? I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers ...
3
votes
1answer
69 views

Set theory, show a set is countable, homework. check my answer

I solved this question but there is something strange going on and I am unsure of myself. Would like someone to review it. We are given a total order (or linear order) $<^{*}$on group $A$ such ...
1
vote
1answer
243 views

zero raised to infinity

I encountered a question where the only condition stated that $t>0$ and was then asked to compare these two quantities $0^t$ $t^0$ The scope of $t$ is $(0,\infty)$ and hence for infinity 1.) ...
4
votes
1answer
62 views

Generlized Büchi Games and Closed under superset Muller Games

For a unique infinite play $p$ in a 2-Player game $G=(V_0,V_1,E)$. Let $$ \inf(p) \subseteq V_0 \cup V_1 $$ be the set of vertices which occur infinitly often in $p$. Generlized B├╝chi (GB) Games ...
1
vote
2answers
1k views

Ratio when one entity is 0.

What is the ratio between boys and girls in a group with 30 boys and 0 girls? Is it 1:0, 30:0 or something involving infinity and undefined? Can somebody help me out here?
3
votes
3answers
148 views

Continuity of $f(x)$ involving infinity

$f(x)= \frac{\sin(\pi x)}{x(1-x)}$ How can I define $f(0)$ and $f(1)$ to make $f(x)$ continuous on $[0,1]$? I've found that the limit at $0 = \pi$, and the limit from the left at $1 = \infty$. I ...
2
votes
5answers
425 views

Prove the set of functions $f : \mathbb{Q} \rightarrow \{1,2,3\}$ uncountably infinite

Prove that the set of functions $f: \mathbb{Q} \rightarrow \{1,2,3\}$ is uncountably infinite. I'm totally stuck on this one. We have just been shown Georg Cantor's diagonalization argument in class ...