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I have different sets of incremental numbers starting from zero like this:

$S_1=\{0,1,2,3\}$

$S_2=\{4,5,6,7\}$

$S_3=\{8,9,10,11\}$

Each set has the same cardinality. I want to know, given a number $x$, the number of sets and the cardinality of the sets, to which set $(S_1,S_2,...)$ does $x$ belong. Is It possible to find it?

Amit Kumar Gupta
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W4lker
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1 Answers1

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Ok, for the sake of generalization I am going to let the starting set be $$S_0$$ $$For\ the\ sets\ S_i\ where\ i,j = 0,1,2,...,n\ and\ s_i \in S_i\ where\ s_i \in \mathbb{N}.$$ $$S_i \cap S_j = \emptyset,\ for\ any\ and\ all\ considered\ sets,$$ $$and\ |S_0| = |S_1| = |S_2|...|S_i| = M\ where\ M\ is\ the\ cardinality\ of\ any\ set\ S_i.$$ $$In\ addition,\ for\ all\ sets,\ S_i\ =\ \left\{i, i+1, i+2,...,i+M - 1\right\}. $$

$$Let\ x \in \mathbb{N},$$$$\ x \in S_i\ if\ and\ only\ if$$ $$x/M = i + k/M\ where\ k\ is\ the\ remainder\ of\ the\ quotient.$$

So, by your example, x will belong to $S_0$ if $x/4\ =\ 0 + j/4,\ where\ 0\ \leq\ j\ < 4.$ Here are some more examples to try and express my point: $$x = 10$$ $$x/4 = 2 + 2/4$$ $$x = 10\ is\ a\ member\ of\ the\ set\ S_2\ as\ i = 2.$$ If there are any suggestions or errors in this argument please comment, I am quite new at this.

Kbroe
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