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10

You've already defined your function (assuming you've also chosen its domain). One of the main ways to "create" a function is simply by specifying its values at all points, and your description has done so. Typical notation for a function created by the sort of description you give is a definition by cases: $$ f(x) := \begin{cases} 0 & x = 0 \\ 1 &...


7

Think about the procedure that you had in mind. If the binary representation of $n$ ends in $1$, then $n$ is odd; say $n=2k+1$. Then $a_n=a_k+1$, and the binary representation of $k$ is simply what’s left when you drop the last digit of $n$. If the binary representation of $n$ ends in $0$, then $n$ is even; say $n=2k$. Then $a_n=a_k-1$, and again the binary ...


4

Actually, you made a mistake breaking down $a_{2015}$. It should be $a_{2015}=a_{2\times 1007+1}=a_{1007}+1$. And $a_{1007}=a_{2\times503+1}=a_{503}+1$ $a_{503}=a_{2\times 251+1}=a_{251}+1$ $a_{251}=a_{2\times 125+1}=a_{125}+1$ $a_{125}=a_{2\times 62+1}=a_{62}+1$ $a_{62}=a_{2\times 31+1}=a_{31}-1$ $a_{31}=a_{2\times 15+1}=a_{15}+1$ $a_{15}=a_{2\times ...


3

In the paper "On nonrepetitive sequences" by Entringer and Jackson, in J. Combin. Theory Ser. A 11 (1974), 159–164. The link is http://www.sciencedirect.com/science/article/pii/0097316574900417, the authors show that any binary sequence of length more than 18 must have two identical consecutive blocks, the length of which is at least 2. The proof is by case ...


2

$C$ is the set of all infinite binary sequences - that is, an element of $C$ looks like $(b_1, b_2, b_3, . . .)$, where each $b_i$ is either $0$ or $1$. So for instance $(0, 1, 0, 1, 0, 1, . . . )$ is an element of $C$, and in this case $b_1=0, b_2=1, b_3=0, . . .$ Re: your first question, you're trying a bit too hard - there's a simpler way. Suppose I have ...


2

How about $f(x)=\left\lceil\frac{x^2}{x^2+1}\right\rceil$ *Works for real numbers, with imaginary numbers you may divide by 0.


1

This would make sense if we interpret $A \cap B$ as the bitwise "and" ($\land$) (or intersection, if you look at sets interpretation), $A \cup B$ as the bitwise "or" ($\lor$) of the numbers (which is their union in the set interpretation). E.g. $7 \cap 11 =$ 0b0111 $\land$ 0b1011 $=$ 0b0011 $= 3$, $7 \cup 11 =$ 0b0111 $\lor$ 0b1011 $=$ 0b1111 $=15 $ , so ...


1

I will assume we are dealing with unsigned integers in binary representation or an boolean algebra which behave like that. i.e. algebra whose elements can be viewed as a sequence of binary digits and multiplication by $2$ corresponds to a shift of the binary digits to the right. i.e. $$z = (z_0, z_1, z_2, \ldots )\quad\implies\quad 2z = ( 0, z_0, z_1, \...


1

If you have an $n$-bit word $w$, flipping all of the bits is the same as subtracting it from $\underbrace{11\cdots 11}_n{}_2$ -- imagine subtracting binary digit for digit; there's never any borrows! Therefore your first procedure computes $$ \underbrace{11\cdots 11}_n{}_2 - (w-1) $$ By a simple algebraic rearrangement, this is the same as $$ \underbrace{11\...


1

I'll answer your last question first since it will give the answer to your other questions. The basic method for establishing that 2 sets have a 1:1 correspondence is constructing a bijection i.e. a one to one and onto function between them. For example, the set of integers $\mathbb{Z}$ is in one to one correspondence with the set of all rationals $\mathbb{Q}...


1

It might be simpler than that, but without knowing exactly what pictures you have in your head, I can't be 100% sure. I think it is a question of remembering that concatenation is essentially multiplication by $x^n$. You take $$R(x) = M(x)\cdot x^n\mod G(x)$$ You say that you "append" $R(x)$ to $M(x)$, which is to say that you calculate $$RM(x) = (M(x)\...


1

I believe your doubt is not really related to CRC but with cyclic codes (on which CRC are based), more specifically with the construction of systematic cyclic codes. This is explained in any textboox. Here's a summary. A binary $(n,k)$ cyclic code has $2^k$ codewords, they correspond to a binary polynomial $c(X)$ of degree less than $n$. A particular code ...


1

You can deliberately define an irrational number to have this non-repeating quality, if required. For example, for an irrational number that avoids 3-blocks of repeating digits, take the binary definition of $\pi$ and generate a new number such that each digit $x$ in $\pi$ is replaced by $11x00$. (This also avoids 4-blocks). To avoid any repetitions of ...



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