Petar Ivanov
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Zeroth homotopy group: what exactly is it?
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14 votes

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 ...

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Find a way out in $8 \times 8$ square
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9 votes

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 ...

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Counterexample or proof of 2-tuple set uniqueness
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6 votes

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 ...

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Let $n ∈ N, n ≥ 1$. Prove that $4^n + 6n - 1$ is divisible by $9$.
4 votes

$$ 4^3 = 64 \equiv 1 \ (mod \ 9)$$ 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 \...

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Show that among all triangles with fixed $s$ and $a$, the area is maximised when $b=c$.
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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 ...

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Find 10th term of sequence based on sum of sequence and 8th term
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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 ...

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What is the value of $\frac{\sin x}x$ at $x=0$?
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 ...

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Surface Area of Rectangular Block
2 votes

The surface area is: $$A = 2*(2x^2 + 2xy + xy) = 4x^2 + 6xy= 4x^2 + \frac{216}{x}$$

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Solve the following complex numbers problem
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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\...

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Does the given vector space form a basis?
2 votes

It doesn't. $R_5[x]$ has dimension 5, while your subset contains only 4 vectors.

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(N,+) and (N,*) aren't isomorphic
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).

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What is $\liminf_{x\to y}f(x)$?
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2 votes

It is $$\lim_{\delta\to0}\quad\inf_{|x-y|<\delta}\quad f(x)$$

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Limit of a sequence
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 \...

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Open Sets: A Real Life Example
1 votes

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 ...

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Reciprocal Polynomial
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1 votes

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

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$A\in M_n$ is nonsingular, $A = BU$, and $U$ is unitary $\Rightarrow $ $A\overline A = B\overline B $
1 votes

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

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Throwing 8 sided dice 3 times with the sum of the throws being 6, whats the probability that the first throw was 3?
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1 votes

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{...

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Probabilty of drawing atleast 1 red ball in 3 attempts.
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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 $(\frac{2}{3})^3 = \frac{8}{27}$. ...

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The distance between two points given distances that bees meet in two different time intervals
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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 ...

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Solution for $\frac{8(9^n)}{8(9^n)+2(4^n)} \gt 0.99$
0 votes

$$\frac{8(9^n)}{8(9^n)+2(4^n)} \gt 0.99$$ $$100(8(9^n)) > 99(8(9^n)+2(4^n))$$ $$8(9^n) > 198(4^n)$$ $$\Big(\frac{4}{9}\Big)^n < \frac{8}{198} = \frac{4}{9}\frac{1}{11}$$ $$\Big(\frac{4}{9}...

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How to determine dimensions of a square based prism
0 votes

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 = ...

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What is the definition of a branch of a function in complex analysis
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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 ...

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How to convert $((x\land y)\lor(z\land u))\land((x\land\neg z)\lor (\neg y \lor u))\land((y\land z)\lor(x\land u))$ to the disjunctive normal form?
0 votes

Hint: Use the following rules multiple times: $$ (a\lor b)\land c = (a\land c)\lor(b\land c)$$ $$ (a\land b)\lor c = (a\lor c)\land(b\lor c)$$

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Find LHS for Induction : Total number of triples selected from N items = N(N-1)(N-2)/6
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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)!}{...

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Prove: If ab is a region in C, then ext ab is open.
0 votes

Start from the definitions: The union of any family of open sets is an open set.

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