Petar Ivanov
• Member for 10 years, 7 months
• Last seen more than 5 years ago
• Redwood City, CA

This is the definition. So $\pi_0$ is the homotopy classes of maps from two points ($S^0$) to $X$, where the first point is mapped to the base point. Clearly only the path connected component matters ...

Imagine this is a chess board, i.e. it's colored in black and white alternatively. Now note that when you make a move you always go from white to black or from black to white. Say the bottom-right ...

Here is a counter example: Family of sets 1: {10, 1, 2} {10, 3, 4} {20, 1, 3} {20, 2, 4} Family of sets 2: {10, 1, 3} {10, 2, 4} {20, 1, 2} {20, 3, 4} This example can easily be extended to ...

$$4^3 = 64 \equiv 1 \ (mod \ 9)$$ So three cases: 1) n = 3k+1: $$4^n+6n-1 \equiv 4+6(3k+1)-1 \equiv 4+6-1 \equiv 0 \ (mod \ 9)$$ 2) n = 3k+2: $$4^n+6n-1 \equiv 16+6(3k+2)-1 \equiv 7+12-1 \equiv 0 \... View answer Accepted answer 3 votes You don't need to differentiate.$$S = \sqrt{s(s-a)(s-b)(s-c)} = C_1\sqrt{(s-b)(s-c)} =  C_1\sqrt{s^2-s(b+c)+bc} = C_1\sqrt{s^2 - s(2s-a)+bc} = C_1\sqrt{C_2+bc}$$And since square root ... View answer Accepted answer 3 votes Let the first term be a and the second a+q. Then the eighth is$$a+7q = 32.5$$and the sum of the first 10 is$$a + a+q + a+2q + ... + a+9q = 10a + 45q = 187.$$Now you can solve this system to ... View answer 2 votes The function is not defined at x=0. But it can be continuously extended (since the limit from the left is equal to the limit from the right). So the graph is not technically correct, it just shows ... View answer 2 votes The surface area is:$$A = 2*(2x^2 + 2xy + xy) = 4x^2 + 6xy= 4x^2 + \frac{216}{x}$$View answer Accepted answer 2 votes It is easy to show that every integer n \in M. i \in M, therefore, 1+i^2=0 \in M. Now applying the same rule we can get arbitrary large integer m \in M. E.g. 0\rightarrow 1\rightarrow 2\... View answer 2 votes It doesn't. R_5[x] has dimension 5, while your subset contains only 4 vectors. View answer 2 votes Let f is the isomorphism. Let f(1)= c. Then using induction f(n) = f(1)^n = c^n, which can't be bijection (since it's not onto). View answer Accepted answer 2 votes It is$$\lim_{\delta\to0}\quad\inf_{|x-y|<\delta}\quad f(x)$$View answer 1 votes We will prove that lim_{n\rightarrow\infty}a_n = 0. We know that$$\forall \epsilon \ \exists N \ \sum_{i=n}^{m}|a_i| < \epsilon, \ n>N, m>N$$. Therefore,$$\forall \epsilon \ \exists N \...

It's important to understand that when talking about topology open set means a set what belongs to the topology. You should't think about it as something 'open' in any other sense. So of course the ...

What they mean is that the new function is called $F$. Such that: $$F(x) = f(\frac{1}{x})$$ Now $$F\left(\frac{1}{r_i}\right) = f\left(\frac{1}{1/r_i}\right) = f(r_i) = 0$$

$$A\bar{A} = BU\overline{BU} = BU\bar{U}\bar{B} = B\bar{B}$$

This is conditional probability. If $A$ is the event that the first throw was $3$ and $B$ is the event that the sum is $6$, then you are looking for $P(A|B)$. Using Bayes' Theorem: $$P(A|B) = \frac{... View answer Accepted answer 1 votes The probability of drawing a non-red ball from one attempt is \frac{10}{15} = \frac{2}{3}. Thus the probability of drawing only non-red balls from three attempts is (\frac{2}{3})^3 = \frac{8}{27}. ... View answer Accepted answer 1 votes Let the slower bee is called A, while the faster bee is called B. Let the whole distance is X. And let they first meet after time t. This means that for time t both bees together travel ... View answer 0 votes$$\frac{8(9^n)}{8(9^n)+2(4^n)} \gt 0.99100(8(9^n)) > 99(8(9^n)+2(4^n))8(9^n) > 198(4^n)\Big(\frac{4}{9}\Big)^n < \frac{8}{198} = \frac{4}{9}\frac{1}{11}\Big(\frac{4}{9}...

If the dimensions of the box are $a$,$b$ and $c$, and the volume is $V$, then you want to minimize $ab+bc+ca$, given that $abc = V$. You can prove that this is achieved when $a=b=c$. Thus, $V = abc = ... View answer 0 votes You can infer the definition: If G is open connected set in$\mathbb{C}$and$f: G \rightarrow \mathbb{C}$is a continuous function such that$z = 1 - f(x)^2$(i.e.$f(x)^2 = 1-z$) then$f$is a ... View answer 0 votes Hint: Use the following rules multiple times: $$(a\lor b)\land c = (a\land c)\lor(b\land c)$$ $$(a\land b)\lor c = (a\lor c)\land(b\lor c)$$ View answer Accepted answer 0 votes Total number of k-tuples selected from n items is the binomial coefficient $$C^k_n = \frac{n!}{k!(n-k)!}$$. In your case$k=3$, so you get:$\$C^3_n = \frac{n!}{3!(n-3)!} = \frac{n(n-1)(n-2)(n-3)!}{...