Partitions of a Set | Set Theory
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- เผยแพร่เมื่อ 12 ก.ย. 2024
- What is a partition of a set? Partitions are very useful in many different areas of mathematics, so it's an important concept to understand. We'll define partitions of sets and give examples in today's lesson!
A partition of a set is basically a way of splitting a set completely into disjoint parts. In other words, if S is a set, then a partition P of S is a collection of nonempty subsets of S such that every element of S is in exactly one element of P. In other words, S is being split into nonempty disjoint subsets such that every element of S is in exactly one of those subsets, and if we put those subsets in a set P - then P is a partition of S.
Lesson on subsets: • What is a Subset?
Lesson on disjoint sets: • What are Disjoint Sets...
I hope you find this video helpful, and be sure to ask any questions down in the comments!
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The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.
Vallow Bandcamp: vallow.bandcam...
Vallow Spotify: open.spotify.c...
Vallow SoundCloud: / benwatts-3
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That was short and straight to the point 🔥 🔥 🔥
Glad it was clear, thanks for watching!
Exactly. Less complicated.
@@brylnicoong5556I am assuming in your ques. It is a^2 - c^2 = d^2 - b^2, although its obvious.
Ans= Equivalence class of (0,0) or [(0,0)] = {(c,d) : (c,d) belongs to RXR(codomain of relation R) and 0^2 - c^2 = d^2 - 0^2}
Now the eq. Is satisfied for real no.s c and d only when c=d=0.
Therefore [(0,0)] = { (0,0) }
Something that my professor doesn't know how to do....
😂@@KP-fd9ev
I really enjoyed the way you explained it, made it easier to understand! Thank you!
So glad it helped! You're welcome and thanks for watching!
Short and straight to the point with simple and concise examples. No overtly complex algebra terminology. Thank you very much for this
Glad to help, thanks for watching!
What a fantastic explanation! It was straight to the point. All I needed was the first couple of minutes. Thank you so much!
the way you make us understand the concept fully, its just amazing, thank you so much
My pleasure - thanks for watching!
Your way of explaining is awesome keep going and we want more videos.
Thank you! More are on the way!
You have explained this important knowledge in a crystal clear way! It is really helpful to me. I will be embracing discrete mathematics this coming semester. And I am super nervous for this. Thank you for your help in helping me understand this point! Thanks!😀
Thank you for making this clear and straight to the point. This made it so much easier to understand
My pleasure, thanks for watching! I highly recommend my three lessons on Bell numbers, a sort of follow up to this lesson - I think they're a lot of fun! Let me know if you ever have any video requests!
Counting Partitions of Sets and Bell Numbers: th-cam.com/video/iJF2kPFGTUo/w-d-xo.html
Bell Numbers and their Recurrence Relation: th-cam.com/video/sPGudyLalmE/w-d-xo.html
The Proof!: th-cam.com/video/abfCpVASfLM/w-d-xo.html
Finally a straightforward and informative explanation! Thank you.
Glad to help!
Thanks mate. Straight to the point
The Partition Theorem was in a “review of probability” document for one of my classes and I swear I had never seen it before lol, so this was really helpful to get me up to speed, thank you lots!
Thanks for this. Very helpful. Might I suggest you leave a little longer between asking the question and giving the answer? I say that because you gave the answer before I could read it all or think, let alone hit the pause button. Just taking my notes below as I go along - if I've misunderstood anything, please feel free to correct in the comments below. Thanks all.
0:14 Basic concept of a partition
0:45 Explanation of why is P a partition of S
*Rules for Partitions*
1:05 Partitions cannot contain empty subsets
1:14 All sets in a partition must be subsets of the 'main' set
1:44 All elements in a set must be included in the partition (and its various subsets)
1:58 No elements in a set can be included in more than 1 subset in the partition
Examples
2:05 Example 1
2:21 Example 2
2:47 Example 3
3:24 Example 4
3:30 Example 5
3:47 Example 6
4:04 Example 7
Explanations
4:36 Re-cap on concepts and terminology
5:20 Full written definition of a partition
5:42 Key point to remember
Thank you! A fine clear understanding better than Zybooks has ever given me!
Awesome, thanks for watching!
The type of explanation I needed. Thank you legend!
Glad it helped, thanks for watching!
@@WrathofMath My textbook has a different definition for partitions, so what do you think is the best way to deal any questions related to partitions?
Two minutes silence for those who choose subtitles 😭💔
setsetsetsetsetsetset
Thank You! Can you please do a video on the recurrence formulation of the Bell number(number of ways to partition a set)?
Thanks for watching, Vishnu! And great idea, I'll get right to work on that lesson! Thanks for the request!
Thank you so much!
THANKS. I am French and in my course we study the link between the number of surjections of n on k and the number of partitions into k blocks of n. But the difference between a surjection and a partition was not explained in the course 💀. But thanks to your video I understand clearly that a partition is equivalent to arranging sets in a set so the order is not important (+ all other constraints obviously) while a surjection amounts to arranging sets in a list so the order is important!
you explanation helped me to understand partitions more clearly than a formal book. thank you for making videos like this
So glad to help!
This video was way better than the shabby uni provided vid thank you
Why can't my professor teach like this , this is so easy thank you for making this so straight forward
So glad it helped, thanks for watching! A few related lessons if you're interested...
Counting Partitions and Bell Numbers: th-cam.com/video/iJF2kPFGTUo/w-d-xo.html
Recurrence Relation for Bell Numbers: th-cam.com/video/sPGudyLalmE/w-d-xo.html
Recurrence proof: th-cam.com/video/abfCpVASfLM/w-d-xo.html
this is just perfect ,loved it thank you
damn!! the best explanation of partition I have ever encountered!! It was clear and to the point!
Absolutely amazing video. Clear explanation and the presentation is 100/10
Thank you!
So much more clear than other videos.
Thanks.
I'm glad it was clear, thanks a lot for watching! Let me know if you ever have any questions!
Smooth lines.. smooth teaching 👌
Thank you!
Definitely one of the best TH-cam channels!
Thanks a lot, I do my best! Let me know if you have any questions, and if you're looking for more on partitions - I have a few more related videos...
Counting Partitions and Bell Numbers: th-cam.com/video/iJF2kPFGTUo/w-d-xo.html
Recurrence Relation for Bell Numbers: th-cam.com/video/sPGudyLalmE/w-d-xo.html
Recurrence proof: th-cam.com/video/abfCpVASfLM/w-d-xo.html
What about for a set like {a,b,c,d} and the partition is {{a,b}, c, {d}} Will this not be a partition because one of the elements is not in brackets which makes it not a subset
Excellent explanation . . easy to understand. . . . . Thank u very much😇
Glad to help, thanks for watching!
You made this look so simple and kinda cool. Thanks from India!
Thanks for watching, so glad it helped!
Nicely described in short time
Thanks Arnav, glad it was helpful! I have several other lessons on partitions and Bell Numbers if you're interested. If you look up "partitions wrath of math" you should find them!
clear and clean af
Thanks for watching!
clear and straight to the point
Glad it was clear, thanks for watching!
Thanks this is really helpful you're very good
So glad it helped! Thanks for watching and let me know if you have any video requests! If you're studying partitions, you may be interested in my lessons on Bell numbers! (just do a search for "Bell numbers wrath of math" and you should find 'em)
Thank you for the explanation
No problem, thanks for watching!
Appreciate this. Do you have a video on partial orders and composition?
Explained it so simply. Much thanks!!
My pleasure, thanks for watching! A few related lessons if you're interested...
Counting Partitions and Bell Numbers: th-cam.com/video/iJF2kPFGTUo/w-d-xo.html
Recurrence Relation for Bell Numbers: th-cam.com/video/sPGudyLalmE/w-d-xo.html
Recurrence proof: th-cam.com/video/abfCpVASfLM/w-d-xo.html
Very nice explanation. Made it easy.
short and straight to the poitn w for you
Fruitful explaining
Glad it helped!
thank you!! you made so easy
Glad to help!
Best explanation.....✨✨🤗
Thank you soo much,, your way of explanation is soo smooth ❤❤
you're the best! this made it so easy to understand
So glad it helped! Thanks for watching and let me know if you ever have any questions!
Clear and concise!
Thank you!
The visual made it clearer 👏🏿
Glad to hear it, thanks for watching!
Thank you very much sir
Is there any formula to calculate how many partition can be done for any number
Excellent explanation! Thank you.
Glad to help!
Straight to the point thank you sir ❤
Glad to help!
Great way of explanation.🇮🇳
Thank you! So glad it was clear!
Thank you!
thank you.. great lecture... I am from bangladesh
Thank you sir
My pleasure!
What cover of moon river is in your outro?
Very good explanation
Thank you, glad you found it clear!
This is usefull!!! Thank u so much
You're welcome!
Great explanation
Just cleared my confusion
Thank you, glad to help!
Thanks
Welcome!
Best explained thank you so much for your effort
Glad it helped! Check out some of my related lessons for more on partitions...
Counting Partitions and Bell Numbers: th-cam.com/video/iJF2kPFGTUo/w-d-xo.html
Recurrence Relation for Bell Numbers: th-cam.com/video/sPGudyLalmE/w-d-xo.html
Recurrence proof: th-cam.com/video/abfCpVASfLM/w-d-xo.html
Thanks a lot sir❤️....
Awesome session...👍
You're very welcome, thank you for watching and let me know if you ever have any questions!
Amazing!
Thank you!
Great! Thanks!
Glad to help - thanks for watching!
Thankyou sir💜
what if the set S has repeating elements , do write it twice in P or do i just follow the definition?
remember a set doesnt contain repeating elements.
Thanks!
No problem!
Any easy formula to find partition of large numbers
Nice explanation
Thanks Debabandan!
Is there any formula that helps us to know the number of partition a set should have? If yes what is it?
Thanks for watching and good question! There is, but it is a recursive formula. Check out my lessons on the Bell Numbers!
Counting Partitions and Bell Numbers: th-cam.com/video/iJF2kPFGTUo/w-d-xo.html
Recurrence Relation for Bell Numbers: th-cam.com/video/sPGudyLalmE/w-d-xo.html
Recurrence proof: th-cam.com/video/abfCpVASfLM/w-d-xo.html
Yes
Thanks
Very useful....🙂
Thank you very much...😃😄
You're very welcome, thanks for watching!
Nice explanation.thank you,sir.👍🏻
My pleasure, thanks for watching!
This was perfect, thanks
My pleasure, glad it helped and thanks for watching!
can i put six numbers in a subset? (like {{1,2,3,4,5,6}}
thank you.
Thank u so much
Most welcome 😊
Thanks :)
You're welcome! :) Thanks for watching!
Love this video keep it up 👍
Thank you very much!
Immediately subscribed after seeing the channel Name. Such a cool Name :D
Haha, thank you! Wrath of Math is the name of a great hip hop album as well, and I love hip hop! If you’re in the mood for some, you may be interested in my latest math rap track: th-cam.com/video/29qzzNEmEOc/w-d-xo.html
THANK YOU SOOOO MUCH
You're very welcome, thanks for watching!
@@WrathofMath I have a question about partitions. { {} , {1,2,3} } is a partition of S ? . In the book the answer is yes but it includes empty. I didnt get it. Thank you so much again.
Good question - I have never seen a definition of partition that allows for empty sets. Partitions must contain only nonempty subsets. If the book says { {} , {1,2,3} } is a partition, it is either a typo or a very nonstandard definition.
@@WrathofMath Okay, thanks again for helping to me
Excellent
Thank you!
TYSM😁
You're welcome! Thank you for watching! I like how your emoji matches Naruto's face in your profile picture!
@@WrathofMath woahhhh, then you're my sensei 😃
Well, frogs and toads happen to be my favorite animals! I’ll be the Math Toad Sage!
@@WrathofMath ohhhhh, Jiraiya Sensei😎
well explained sir
Thank you!
THANK YOU
You're welcome! Thank you for watching!
what if (1,3) (2), (1,3), is it still partition?
No, think of it like you can take every value from the set and separate them into different sets even together. But once you add number outside your set space like, let's say you add 4, it's no longer a partition of the set. And if we add extra values into the set like you did here, it's no longer a partition. You would need {1,1,2,3,3} as your set if you wanted that to be a partiton. Im assuming you are going off the set {1,2,3}, which would make what you have not a partions as it adds the extra values into the mix. Hoefully, this helps some.
plz upload the video for cross partition
Noooowwwww I get it. Thank you!
Glad it helped and thanks for watching!
Thanks😊
You're very welcome! 😊 Thanks for watching!
In How many ways can we do partition of a set?
Plz..reply sir
Plz reply sir🙏🏻
Depends on how big the set is
Only as many ways as the number of elements in the set you want to partition
Great sir
Thank you!
Awesome
Thank you!
What is the formula for finding partition of a set
Thanks for watching, and do you mean a formula for counting the number of partitions of a set? Check out this sequence of 3 lessons, ending with a proof of a recurrence relation that can be used to calculate the number of partitions of a set (the number of partitions of a set with n elements is called a Bell number, B_n).
th-cam.com/video/iJF2kPFGTUo/w-d-xo.html
th-cam.com/video/sPGudyLalmE/w-d-xo.html
th-cam.com/video/abfCpVASfLM/w-d-xo.html
Youre a beast
I do my best - thanks for watching!
studying in Germany, being russian, watching math explanation in english
thanks
That's incredible! Thanks for watching, so glad to help!
2:06 I thought the third one contains 3...
nice
6:09. Green elements...🤨😦
please come teach in our university please
Awesome!
Love how you gave up on drawing curly brackets correctly 😂
good
Thank you!
Sir please hindi and English mix karke is questions Ko batai n