1

Let $A=\{10,101\}$ and $B=\{001,100,1001\}$.Write $AB$ and $BA$ ,determine if they are uniquely created and their generating functions with respect to length.

If $A$ and $B$ are sets of binary string, then $AB= \{ab \}$ where $a$ is in $A$ and $b$ in $B$.

Do you have any idea for generating function ? Or uniqueness part.

Thank you.

mathsstudent
  • 181
  • There are only $2\times 3 = 6$ possible combinations in $AB$, so it is trivial to check them for uniqueness. – Paul Sinclair Mar 22 '19 at 02:01
  • Yes. I checked. They are uniquelly generated. But I do not have any idea for generating function. Do you have any this part ? – mathsstudent Mar 22 '19 at 18:48
  • "10" + "1001" = "101" + "001". That is not uniquely generated. And I'm sorry, but I don't even know what you would consider a "generating function" for this. – Paul Sinclair Mar 22 '19 at 20:18

1 Answers1

0

We obtain \begin{align*} AB&=\{10,101\}\{001,100,1001\}\\ &=\{10001,10100,\color{blue}{101001},\color{blue}{101001},101100,1011001\}\\ &=\{10001,10100,101001,101100,1011001\}\\ \end{align*} with generating function \begin{align*} \Phi_{AB}(x)=2x^5+2x^6+x^7 \end{align*}

We see $AB$ is not uniquely created due to $10\cdot1001=101\cdot 001$. The coefficient $[x^n]$ of the generating function $\Phi_{AB}$ gives the number of words of length $n$.

We obtain \begin{align*} BA&=\{001,100,1001\}\{10,101\}\\ &=\{00110,001101,10010,100101,100110,1001101\} \end{align*} with generating function
\begin{align*} \Phi_{BA}(x)=2x^5+3x^6+x^7 \end{align*}

We see $BA$ is uniquely created and we consequently have $AB\neq BA$. We also have $\Phi_{AB}(x)\neq \Phi_{BA}(x)$.

Markus Scheuer
  • 108,315