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I have a product of the form:

$$ a_{1}^\alpha a_{2}^\beta a_{3}^\gamma ... a_{k}^\zeta $$

where each of $$\alpha,\beta,\gamma,\delta,...\zeta$$ can take values from 0 to n

$$ \alpha,\beta,\gamma,\delta,...,\zeta \in \{0, 1, 2, 3, ... , n \} $$

I'm using alpha, beta, .. because I have trouble with double subscripts otherwise I'd use \alpha_{1},...,\alpha_{n+1}

and $$a_{1}, a_{2}, ...,a_{k} $$ are coprime.

With other words the product has a variable length of elements and the set S contains all products resulted of the factors with permutated powers...

Now my first question is, how can mathematicy represent the set S?

$$ S = \{a_{1}^0a_{2}^0a_{3}^0...a_{k}^1,a_{0}^0a_{2}^0a_{3}^0...a_{k}^1,...,a_{1}^1a_{2}^0a_{3}^0...a_{k}^0,...,a_{1}^na_{2}^na_{3}^n...a_{k}^n\} $$

My second question is, how would I calculate the cardinality of this set?

1 Answers 1

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You can define the set $S$ as

$$ S=\left\{\prod_{i=1}^ka_i^{\alpha_i}: \alpha_i\in\{0,1,\dots,n\}\right\} $$

As far as the cardinality of $S$ is concerned, you have $n+1$ possibilities for each of the $k$ terms $a_1,\dots a_k$ (since $n+1$ is the cardinality of $\{0,1,\dots n\}$). Since $a_i,\dots a_k$ are coprime, no element can be counted twice. Hence $$|S|=(n+1)^k.$$

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    It's not so simple. Since the bases are not necessarily coprime, some choices of exponents can result in the same value. – 2017-02-16
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    I interpreted the question more as a request of counting all the elements of $S$ regardless of whether two elements can be equal or not. Your objection is perfectly right in your interpretation, but then it's impossible to answer without knowing anything about $a,b,c,d$. – 2017-02-16
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    I don't know about impossible without explicit values, but you are right that it does not look easy. – 2017-02-16
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    Of course it is possible to list a series of cases with different assumptions on $a,b,c,d$, but it would be an extremely long (possibly infinite) list different cases. Thanks for pointing it out anyway – 2017-02-16
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    Well, that would be the most obvious approach, but necessarily the only one. – 2017-02-16
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    Given a number of terms in the product a,b,c,d, .. let's rather call them $$a_{k}$$ for convenience. These elements are coprime indeed, I didn't specify that, thanks for pointing out. I would then have to represent the terms in S using product notation over unique permutations of alpha, beta, (this is mainly my difficulty) ... and the cardinality of |S| will be given by (n+1)^k if I understand this correctly. How could this be then represented? – 2017-02-16
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    @alle2004 If the elements are coprime, then you should edit the question. – 2017-02-16
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    @alle2004 good to hear that Qudit's remark was useful. I suggest you edit your question to include the fact that $a,b,c,d$ are indeed coprime. What do you mean by "how could this be represented"? – 2017-02-16
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    @alle2004 Sorry to be picky, but if $a,b,c,d$ are coprime, my answer is correct. Why did you un-accepted it? – 2017-02-16
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    @alle2004 I saw your edit to the question and re-answered the question. However, I suggest you edit the question again: it doesn't make sense to take $\alpha,\beta,\gamma,\delta$ now since you have $k$ elements (not only 4) – 2017-02-17
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    @alle2004 You shouldn't edit a question if it substantially changes it. It is better to ask a separate question. – 2017-02-17
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    @John, just to clarify, using your notation, will alpha take (potentially) different values for each factor or will it take the same value alpha_{i} for all factors in all products? – 2017-02-17
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    @alle2004 of course it can take different values. Each $\alpha_i$ can be different from the others. – 2017-02-17