Then I was studying at 11th std when we were being introduced to 'Sets' my maths tutor at P.C.Thomas Classes told me that there was no 'real' formula for finding the number of partitions for a set of $x$ elements.But then I came across the Bell number which gave the number of partitions as a recurrence series.
But still I was not satisfied, Bell number was not really a formula and there was no explanation why it represent the number of partitions.
Nearly a year passed by, I learned many new things which included functions. Suddenly I had a insight that Surjective funtions beared a close relationship with set partitions.
Finally I arrived a the summation formula:
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Proof:
A function $f:X->Y$ is said to be surjective if every element of $Y$ is a image of some element of $X$ .
($x$ & $y$ are respectively the number of elements in $X$ & $Y$).$\left ( x\geq y \right )$
But still I was not satisfied, Bell number was not really a formula and there was no explanation why it represent the number of partitions.
Nearly a year passed by, I learned many new things which included functions. Suddenly I had a insight that Surjective funtions beared a close relationship with set partitions.
Finally I arrived a the summation formula:
Where $x$ is the number of elements in the set.
Proof:
A function $f:X->Y$ is said to be surjective if every element of $Y$ is a image of some element of $X$ .
($x$ & $y$ are respectively the number of elements in $X$ & $Y$).$\left ( x\geq y \right )$
