# ðŸ”” Bell Numbers and Its Recurrence Relation (Proof)

Share
Embed
• Published on Sep 23, 2018
• We define Bell Numbers and derive its recurrence relation.
For exponential generating function: usclip.net/video/1I4VpemnWmg/video.html
Your support is truly a huge encouragement.
Please take a second to subscribe in order to send us your valuable support and receive notifications for new videos!
Every subscriber and every like are wholeheartedly appreciated.

## Comments • 14

• Sumit Kumar 4 months ago

Excellent.

• Duvvada Jayanth Kumar 8 months ago +1

Tq very much

• James Wilson 9 months ago +1

Sweet! I love this kind of stuff!

• Vighnesh Raut 9 months ago +1

When you said partitions, I was thinking about the partitions that Ramanujan generalised. But these are different.

• Konrad KamiÅ„ski 9 months ago +3

According to given definition B(0) =0 there is no partitioning of empty set without using empty set, but theorem works only when we set B(0)=1.

• Konrad KamiÅ„ski 9 months ago

Or you can say that partition is maximal set of disjoint subsets. It doesn't change B for n greater than 0 but it defines B(0)=1 correctly. But then we have to live with every partition containing empty set. ;) Ehh, life is hard!

• LetsSolveMathProblems  9 months ago +3

One may argue that we can consider {empty set} as a "partition" of empty set, but you are right--from the definition I gave, B(0) = 0. Perhaps I should have mentioned that our definition applies only for positive integer n, then defined B(0) as 1 at the beginning of the video.

• Logan Hargrove 9 months ago +3

Thank you! I have been working on a certain infinite series and I just now realized that the result can be expressed in terms of Bell numbers.

• Paco Gomez-Paz 9 months ago

I am trying to prove that the cardinality of a bell set is greater than that of the original set for both infinite and non infinite sets non infinite is quite easy to prove but I am having troubles for infinite sets . Anyone have any ideas

• Paco Gomez-Paz 8 months ago

I meant greater than or equal to also I already proved it but thank you for taking the time to respond

• Konrad KamiÅ„ski 9 months ago

What is Bell set? Is it set of all partitions? Then it's not true for n=1? For infinite just notice that Bell set's power is greater or equal than Power set's cardinality (using partitions of 2 sets) and Power set's cardinality is strictly greater than cardinality of original set.
If Bell's set is something different, plz tell your definition.
PS. Actually now I see that B(Kappa)=2^(Kappa) for Kappa greater or equal Aleph_0.

• adandap 9 months ago +6

Nice, though essentially an induction proof.

• Xander Gouws 9 months ago +11

Your vids are really warming me up to combinatorics

• GENIUS 9 months ago

Awesome!!!!!!I absolutely love your videos!!!Keep them coming.Oh,and by the way,could you solve this integral that's been bugging me:int from 0 to pi/4 of(1-(x)^2+(x)^4-(x)^6...)/((cosx)^2+(cosx)^4+(cosx)^6+....).You can find it here around the 1:34:00 mark in the integration bee here: usclip.net/video/UNpa3-EGxGY/video.html