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Registered: 2009-10-08

Posts: 6

"Symbols vs words of length 1" OR "picking nits"

I just got a decent discrete math text and am learning about sets.  They've just introduced "alphabets" as a type of set that contains "symbols".  No definition is given for a symbol, which is interesting.  It is interesting because the following could be seen as a set of sets, or as an alphabet:

Of course, it depends on whether A, B, and C are sets or symbols.  This is strange because, if I am not mistaken, math is written in symbols.

Next the text defines a "word" or "string" as an ordering of symbols.

So words are constructed by placing symbols in ordered sequences.

My question is subtle, I hope you won't mistake it for dumb:

Is a symbol a word of length 1, or is it not a word at all?

Why does it matter?  Well, I'm wondering if a set of words of length 1 is an alphabet.  Perhaps symbols are not words.  There is a difference, after all, between an ordered sequence containing one symbol, and a symbol, is there not?

Think of it this way.  Imagine that to create a word, the notation was a little more precise, and we had to use, say braces around the symbols.  Then we could make words like

and the null word,

The null word is an object because it is an ordered list of symbols containing zero symbols, just like the null set is an unordered collection of zero elements.  On this basis, the word

should be different from the symbol

just as the set

is different from the number.

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Registered: 2009-10-08

Posts: 6

Re: "Symbols vs words of length 1" OR "picking nits"

Not sure if it came across right: this is a question.  I'm sure someone out there has an opinion at least?

So the question is "Are a symbol and a word of length 1 equivalent".  I proposed my answer "no" and gave my reasons above.  Do you agree that a symbol and a 1-word are different objects?

mathsyperson

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Registered: 2005-06-22

Posts: 4,900

Re: "Symbols vs words of length 1" OR "picking nits"

I read the question and thought no, before scolling down and reading pretty much the reasoning I had just gone through.

Words are made by stringing letters together. You can't combine words together to make new words. Therefore, letters are not automatically length-1 words.

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Why did the vector cross the road?
It wanted to be normal.

listening

Member

Registered: 2009-10-08

Posts: 6

Re: "Symbols vs words of length 1" OR "picking nits"

Good idea!  Can you build other words from the 1-words?

Since the whole idea of symbols is to make words, if the 1-words cannot be used to make words, we know that it doesn't make sense to think of the set of all 1-words over an alphabet as the same as the alphabet.  Oh, this lets me formalize my question a different way:

This is an interesting way to state that symbols and letters are the same thing.  I wonder if I could prove that statement false...
I'm not very comfortable with proving.

In any case, back to your idea: we could try to make the language ∑* using only the set of 1-words S defined above.  So you say that words cannot be combined together to make other words?  Not with that attitude!  In fact, we can define the operation concatenation, as suggested by my textbook, which creates a new word that has the symbols of two other words.

In that case, it seems we could create all words, and indeed assemble ∑* by using concatenation of words from S.

Is there any problem with that?... yes.  In fact there are some words we cannot make by that proposed method:
- we can't make the null word,
- nor any 1-words.

At least this set of words of length < 2 are missing.  Of course, if we added to S the null word, then we could make the null word by concatenating it with itself, and 1-words by concatenating the desired 1-word with the null word.

It seems like 1-words are different then symbols.  I'd still like to formally prove it, just to flex a bit, but I don't even really understand what rules are in terms of what a "formal proof is".

dannyv

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Registered: 2007-09-20

Posts: 34

Re: "Symbols vs words of length 1" OR "picking nits"

I read the question and thought no, before scolling down and reading pretty much the reasoning I had just gone through.

Words are made by stringing letters together. You can't combine words together to make new words. Therefore, letters are not automatically length-1 words.

Actually, languages have operations defined on it, like concatenation, union, transitive closure, and others, but those are the basics. Alphabets, these operators and some rules makes a really nice algebra.

Last edited by dannyv (2009-10-11 01:48:45)

dannyv

Member

Registered: 2007-09-20

Posts: 34

Re: "Symbols vs words of length 1" OR "picking nits"

Good idea!  Can you build other words from the 1-words?

This is an interesting way to state that symbols and letters are the same thing.  I wonder if I could prove that statement false...
I'm not very comfortable with proving.

You don't need to prove that, the difference here is that Sigma is an alphabet and S is a language. Just remember that Strings/Word of lenght 1 are not symbols, and symbols are not strings of lenght 1.