Exploring the Properties of \Sigma: \beta, x, y, z

In summary, the symbols in \Sigma are defined as \beta, x, y, and z, where \beta represents a blank. The operators \parallel and \lambday are used to compute the length of a string and the number of symbols in a string, respectively. The empty string \lambda is equivalent to the string with no symbols from \Sigma. From this, it can be determined that the length of \lambda is 0 and the length of \beta is 1. Thus, the length of \beta^3 is 3 and the length of \lambda^{10} is 0.
  • #1
prevail
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Let [tex]\Sigma[/tex] = {[tex] \beta[/tex],x,y,z} where [tex] \beta [/tex] denotes a blank, so x[tex]\beta \neq[/tex] x, [tex]\beta \beta \neq \beta[/tex], and x[tex]\beta[/tex]y [tex]\neq[/tex] xy but x [tex] \lambda[/tex]y = xy.

Compute each of the following:

1: [tex] \parallel \lambda \parallel [/tex]
2: [tex] \parallel \lambda \lambda \parallel [/tex]
3: [tex] \parallel \beta \parallel [/tex]
4: [tex] \parallel \beta \beta \parallel [/tex]
5: [tex] \parallel \beta[/tex]3 [tex] \parallel [/tex]
6: [tex] \parallel[/tex] x [tex] \beta \beta [/tex] x [tex] \parallel [/tex]
7: [tex] \parallel \beta \lambda \parallel [/tex]
8: [tex] \parallel \lambda [/tex] 10 [tex] \parallel [/tex]

Uhm.. can someone help me out ? :cry: I've tried like 3 days now (without progress).
 
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  • #2
What have you done so far?
 
  • #3
Tom Mattson said:
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.. :cry:

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

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.. :smile:
 
  • #4
I'm surprised that nobody can solve this ...
 
  • #5
You haven't given a whole lot of information! You said [itex]\beta[/itex] represents a blank (I guess we might call that a "hard" blank) so really is treated just as another symbol. But what is [itex]\lambda[/itex]? The only thing you tell us about that is "but x[itex]\lambda[/itex]y= xy". So [itex]\lambda[/itex] is a "soft" blank- like nothing? Is [itex]\beta^3[/itex] the same as [itex]\beta\beta\beta[/itex]? And what, exactly is the definition of [itex]\parallel \parallel[/itex]? It would guess it is the length of the string but it would be a good idea to say that explicitely.
 
  • #6
HallsofIvy said:
You haven't given a whole lot of information! You said [itex]\beta[/itex] represents a blank (I guess we might call that a "hard" blank) so really is treated just as another symbol. But what is [itex]\lambda[/itex]? The only thing you tell us about that is "but x[itex]\lambda[/itex]y= xy". So [itex]\lambda[/itex] is a "soft" blank- like nothing? Is [itex]\beta^3[/itex] the same as [itex]\beta\beta\beta[/itex]? And what, exactly is the definition of [itex]\parallel \parallel[/itex]? It would guess it is the length of the string but it would be a good idea to say that explicitely.

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

[tex] \{ \lambda \} \neq \emptyset [/tex] because [tex]| \{ \lambda \} | = [/tex] 1 [tex] \neq[/tex] 0 [tex] = | \emptyset | [/tex].

[tex]\parallel [/tex] w [tex]\parallel[/itex] = the length of w, and [tex] \parallel \lambda \parallel [/tex] = 0. [tex]\parallel \beta \parallel [/tex] = 1. ...

Sorry for the lack of information.. :frown:
 
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  • #7
Some of them are obvious...
 
  • #8
Indeed all of them are obvious. It's just counting!
1.[tex] \parallel \lambda \parallel= 0 [/tex]

2.[tex] \parallel \lambda \lambda\parallel= 0 [/tex]

3.[tex] \parallel \beta \parallel= 1 [/tex]

4.[tex] \parallel \beta^3= 3[/tex]

.
.
.
8. [tex] \parallel \lambda^{10}= 0 [/tex]
 
  • #9
HallsofIvy said:
Indeed all of them are obvious. It's just counting!
1.[tex] \parallel \lambda \parallel= 0 [/tex]

2.[tex] \parallel \lambda \lambda\parallel= 0 [/tex]

3.[tex] \parallel \beta \parallel= 1 [/tex]

4.[tex] \parallel \beta^3= 3[/tex]

.
.
.
8. [tex] \parallel \lambda^{10}= 0 [/tex]

So.. nr 6 is like.. 4, right?
 
  • #10
Yes, that is correct- just count the number of symbols in the string.
 
  • #11
aight, thanks for the help dude
 

What is \Sigma?

\Sigma is the Greek letter sigma, which is commonly used in mathematics and science to represent summation or a sum of values.

What are the properties of \Sigma?

Some of the properties of \Sigma include linearity, distributivity, and commutativity. Linearity means that the sigma symbol can be moved inside or outside of a function without changing the value. Distributivity means that the sigma symbol can be distributed over addition or subtraction. Commutativity means that the order of the terms being summed does not affect the final result.

What do \beta, x, y, and z represent in the context of \Sigma?

In the context of \Sigma, \beta, x, y, and z can represent variables or values that are being summed together. These variables can represent any type of data, such as numbers, vectors, or matrices.

How is \Sigma used in scientific research?

\Sigma is commonly used in scientific research to represent the summation of data or values. It can be used to calculate averages, totals, or other statistical measures. It is also used in mathematical proofs and equations to represent a series of values.

Are there any limitations to using \Sigma?

One limitation of using \Sigma is that it can only be used for discrete data, meaning data that is countable or can be divided into distinct categories. It cannot be used for continuous data, such as measurements on a scale. Additionally, the order of the terms being summed can affect the result, so it is important to be careful when rearranging terms.

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