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I am trying to a formal expression for a set $S$ of $T$ number of ones.

$ S = \{1,1,1,1,....T$ times $\}$.

Currently I have written it in this way:

$ S = \{n_i | n_i = 1, \forall i \in [1,T] \}$.

Is this expression correct? Are there better ways of writing the same thing?

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There exists a notion of multisets, maybe it might be useful for you? –  Martin Sleziak May 23 '12 at 11:06
    
yes multisets can be useful. will look into them. Thank you –  maths-help-seeker May 23 '12 at 11:36

3 Answers 3

up vote 3 down vote accepted

In set theory, the naive definition of a set is "a collection of objects without an order or repetition". This means that $\{1\}=\{1,1\}=\{1,1,1,1,1,\ldots\}$ as well $\{1,2\}=\{2,1\}$.

There is a concept called multi-set in which order is of no importance, but repetition counts. I have never used these and I am not sure what would be the correct notation for them.

Lastly, there are also sequences (or ordered sets) in which both the order and the repetition are important, $\langle 1,1,1\rangle\neq\langle 1,1\rangle$ and $\langle 1,2\rangle\neq\langle 2,1\rangle$.

One final remark is that whenever you have an object you wish to work with the important thing is to define it clearly, be consistent with its notation and do your best to avoid overloading previously used symbols.

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thank you for the ideas! I will look into both multisets and sequences. –  maths-help-seeker May 23 '12 at 11:37
    
Infact, in my case, sequence is the most appropriate fit. I want them to be in the same order. –  maths-help-seeker May 23 '12 at 11:54
    
@maths-help-seeker: I am glad to have helped. –  Asaf Karagila May 23 '12 at 11:59

Your suggestion would just give you the set $\{1\}$ by extensionality. Think of it as this: the sets $\{x,x\}$ and $\{x\}$ have the same elements, therefore they are equal.

But you still can get a sequence of $T$ ones if you want. Look at the set $\mathbb{N}^T$, which consists of $T$-tuples of natural numbers, or functions from the set $\{0,\ldots,T-1\}$ to $\mathbb{N}$. The function which is constant $1$ would probably do what you want.

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A set is defined by its unique members: $\{1,1\}$ and $\{1\}$ are simply two different names for the same set, the set whose only member is the number $1$. If you want to have something set-like that allows multiple copies of its members, you want a multiset. I’m not aware of any completely standard notation for multisets.

One possibility, however, is basically what you’ve done: write the multiset as an indexed family. I’d handle the details a little differently, however, especially if you want to be able to deal with different values of $T$ at once. For each $i\in\mathbb Z^+$ let $n_i=1$. Then for any particular $T$ you can define $S_T=\{n_i:i=1,\dots,T\}$.

If you use the formal definition of multiset given in the Wikipedia article, your $S$ is $\Big\langle \{1\},\{\langle 1,T\rangle\}\Big\rangle$.

And perhaps the simplest possibility is just to define $S_T$ in words, as the multiset of $T$ $1$’s. Clarity is usually preferable to formality, if a choice is to be made.

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