# All Questions

479 views

### Injectivity and Surjectivity of a piecewise defined function

Let there be a function $f\colon\def\R{\mathbb R}\R \to \R$ given by $$f(x)= \begin{cases} 5x+2 &x\ge 1 \\ x-1 &x<1 \end{cases}, \qquad x \in \R.$$ Prove that $f$ is not surjective, also ...
90 views

### comparing bit lengths of binary numbers

Suppose I have two binary numbers x and y that have bit lengths of nx and ...
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### the division $14^{256}$ by $17$ [duplicate]

What the rest of the division $14^{256}$ by $17$? $$14^2\equiv9\pmod{17}\\14^4\equiv13\pmod{17}\\14^8\equiv16\equiv-1\pmod{17}\\(14^8)^{32}\equiv(-1)^{32}\equiv1\pmod{17}$$The rest is $1$, ...
92 views

### Functional equation, find functions

Find all of the functions defined on the set of integers and receiving the integers value, satisfying the condition: $$f(a+b)^3-f(a)^3-f(b)^3=3f(a)f(b)f(a+b)$$ for each pair of integers $(a,b)$.
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### Any 'odd unit fraction' whose denominator is not $1$ can be represented as the sum of three different 'odd unit fractions'?

Let us call a fraction whose denominator is odd 'odd fraction'. Also, let us call an odd fraction whose numerator is 1 'odd unit fraction'. Then, here is my question. Question : Is the following ...
1k views

Given the function: $$v(x,y) = x + e^{-((x-1)^2 + (y-1)^2)}$$ I am supposed to calculate the gradient of this expression in Matlab for x defined in the interval -1:0.1:0.9 and y defined in the ...
59 views

### Congruences doubt!

What the rest of the division $2^{100}$ by $11$? $$2^5=32\equiv10\equiv-1\pmod{11}\\(2^5)^{20}=2^{100}\equiv-1^{20}\;\text{or}\; (-1)^{20}$$??
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### Prove the following statement…

So, how can I prove this limit: $$\lim_{x \to x_0} \frac{1}{(x-x_0)^2}=+\infty$$ What I've tried is to multiply both numerator and denominator with (x+x0), but I guess that's wrong, what can I do to ...
59 views

### prove $| \cup_n A_n| <\mathfrak{c}$

If $\{A_n: n \in \mathbb{N} \}$ is a sequence of subsets of $\mathbb{R}$ and $|A_n| < \mathfrak{c}$ for all $n$. Prove $| \cup_n A_n| <\mathfrak{c}$ with $\mathfrak{c}$ the cardinality of ...
100 views

### Find pattern in recursion

this is a example from my book Write several elements of the recursion, and see if you can find a pattern. T(n) = T(n – 1) + n T(n –1) = T(n – 2) + (n –1) T(n –2) = T(n – 3) + (n –2) T(n –3) ...
116 views

### Hypergeometric Function Differential Equation

Is there some nice obvious way to see that the hypergeometric function $$_2F_1(a,b;c:z) = \sum_{i=0}^\infty \tfrac{(a)_n(b)_n}{(c)_n}\tfrac{z^n}{n!}$$ should satisfy the differential equation ...
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### Discrete and combinatorial mathematics

Suppose we have a relation on a set $A$, i.e. $A \times A$, where $|A| = n; \;n$ a positive integer. How can we count the number of relations on set $A$ which are reflexive, symmetric, transitive and ...
87 views

### marching band conductor

Let $f(x)$ be the unique polynomial that satisfies: $f(n)=\sum_{i=1}^{n} i^{101}$, for all positive integers $n$. The leading coefficient of $f(n)$ can be expressed as $\frac {a}{b}$, where $a$ and ...
32 views

### Find an infinite set?

$X$ is the union of the intervals $[\frac{1}{n^2},2-\frac{3}{\sqrt{n}}]$ for $n=1$ to infinity. Find $X$. My question is about the first few intervals in the union. When $n=1$, the interval is ...
146 views

### When f is absolutely integrable and contiunous prove that $\sqrt{f}$ is absolutely integrable.

If $f: (a ,b) \rightarrow [0,\infty)$ is continuous and absolutely integrable on (a,b), then prove that $\sqrt{f}$ is absolutely integrable on (a,b). I have that $\sqrt{f}$ is locally integrable. I ...
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### Proof that $P(\cup_{n=1}^\infty\cap_{m=n}^{\infty}A_m^c)=1$

Let $A_1,A_2,...$ be events and P be probability measure. If $\Sigma_{n=1}^{\infty}P(A_n)<\infty$ then prove that $P(\cup_{n=1}^\infty\cap_{m=n}^{\infty}A_m^c)=1$ ie. ...