Computing string lengths with blank symbol β in Σ

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Let $$\Sigma$$ = {$$\beta$$,x,y,z} where $$\beta$$ denotes a blank, so x$$\beta \neq$$ x, $$\beta \beta \neq \beta$$, and x$$\beta$$y $$\neq$$ xy but x $$\lambda$$y = xy.

Compute each of the following:

1: $$\parallel \lambda \parallel$$
2: $$\parallel \lambda \lambda \parallel$$
3: $$\parallel \beta \parallel$$
4: $$\parallel \beta \beta \parallel$$
5: $$\parallel \beta$$³ $$\parallel$$
6: $$\parallel$$ x $$\beta \beta$$ x $$\parallel$$
7: $$\parallel \beta \lambda \parallel$$
8: $$\parallel \lambda$$ ¹⁰ $$\parallel$$

Uhm.. can someone help me out ? I've tried like 3 days now (without progress).

 

[quantumdude]

What have you done so far?

 

[prevail]

What have you done so far?

Well.. the problem is that I'm totally stuck. I have no idea what to do.. I've red the chapter over and over, checked several math websites, forum and so on..

It seems to me that people find it difficult to solve this no matter math skills

So if you don't want to help me (the assigment was handed in today).. that's ok. I can go on not understanding this..

 

[prevail]

I'm surprised that nobody can solve this ...

 

[HallsofIvy]

You haven't given a whole lot of information! You said $\beta$ represents a blank (I guess we might call that a "hard" blank) so really is treated just as another symbol. But what is $\lambda$? The only thing you tell us about that is "but x$\lambda$y= xy". So $\lambda$ is a "soft" blank- like nothing? Is $\beta^3$ the same as $\beta\beta\beta$? And what, exactly is the definition of $\parallel \parallel$? It would guess it is the length of the string but it would be a good idea to say that explicitely.

 

[prevail]

You haven't given a whole lot of information! You said $\beta$ represents a blank (I guess we might call that a "hard" blank) so really is treated just as another symbol. But what is $\lambda$? The only thing you tell us about that is "but x$\lambda$y= xy". So $\lambda$ is a "soft" blank- like nothing? Is $\beta^3$ the same as $\beta\beta\beta$? And what, exactly is the definition of $\parallel \parallel$? It would guess it is the length of the string but it would be a good idea to say that explicitely.

$$\lambda$$ is according to definition a empty string - that is, the string consisting of no symbols taken from $$\Sigma$$.

$$\{ \lambda \} \neq \emptyset$$ because $$| \{ \lambda \} | =$$ 1 $$\neq$$ 0 $$= | \emptyset |$$.

$$\parallel$$ w [tex]\parallel[/itex] = the length of w, and $$\parallel \lambda \parallel$$ = 0. $$\parallel \beta \parallel$$ = 1. ... <br /> <br /> Sorry for the lack of information.. <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f641.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":frown:" title="Frown :frown:" data-smilie="3"data-shortname=":frown:" />[/tex]

 

[prevail]

Some of them are obvious...

 

[HallsofIvy]

Indeed all of them are obvious. It's just counting!
1.$$\parallel \lambda \parallel= 0$$

2.$$\parallel \lambda \lambda\parallel= 0$$

3.$$\parallel \beta \parallel= 1$$

4.$$\parallel \beta^3= 3$$

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8. $$\parallel \lambda^{10}= 0$$

 

[prevail]

Indeed all of them are obvious. It's just counting!
1.$$\parallel \lambda \parallel= 0$$

2.$$\parallel \lambda \lambda\parallel= 0$$

3.$$\parallel \beta \parallel= 1$$

4.$$\parallel \beta^3= 3$$

.
.
.
8. $$\parallel \lambda^{10}= 0$$

So.. nr 6 is like.. 4, right?

 

[HallsofIvy]

Yes, that is correct- just count the number of symbols in the string.

 

[prevail]

aight, thanks for the help dude