Partitions of a Set | Set Theory

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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[@snrikhotso]

That was short and straight to the point 🔥 🔥 🔥

[@WrathofMath]

Glad it was clear, thanks for watching!

[@brylnicoong5556]

Exactly. Less complicated.

[@ritikraturi2560]

@@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) }

[@KP-fd9ev]

Something that my professor doesn't know how to do....

[@blackangelinyourarrea]

😂​@@KP-fd9ev

[@oleevjen-caspersen6200]

I really enjoyed the way you explained it, made it easier to understand! Thank you!

[@WrathofMath]

So glad it helped! You're welcome and thanks for watching!

[@chukwukelumarvelous9616]

Short and straight to the point with simple and concise examples. No overtly complex algebra terminology. Thank you very much for this

[@WrathofMath]

Glad to help, thanks for watching!

[@kyusiv9026]

the way you make us understand the concept fully, its just amazing, thank you so much

[@WrathofMath]

My pleasure - thanks for watching!

[@grifo_-cyan7123]

What a fantastic explanation! It was straight to the point. All I needed was the first couple of minutes. Thank you so much!

[@sharmilap2468]

Your way of explaining is awesome keep going and we want more videos.

[@WrathofMath]

Thank you! More are on the way!

[@EmmanuelTamenut]

Finally a straightforward and informative explanation! Thank you.

[@WrathofMath]

Glad to help!

[@jingyiwang5113]

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!😀

[@erzphantomhive1]

Thank you for making this clear and straight to the point. This made it so much easier to understand

[@WrathofMath]

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: ua-cam.com/video/iJF2kPFGTUo/v-deo.html
Bell Numbers and their Recurrence Relation: ua-cam.com/video/sPGudyLalmE/v-deo.html
The Proof!: ua-cam.com/video/abfCpVASfLM/v-deo.html

[@jsc0625]

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!

[@latedeveloper7836]

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

[@mishrashubham007]

Thanks mate. Straight to the point

[@jakewatson3369]

Thank you! A fine clear understanding better than Zybooks has ever given me!

[@WrathofMath]

Awesome, thanks for watching!

[@பேராண்டி]

Two minutes silence for those who choose subtitles 😭💔

[@WrathofMath]

setsetsetsetsetsetset

[@nazmiepatel720]

The type of explanation I needed. Thank you legend!

[@WrathofMath]

Glad it helped, thanks for watching!

[@nazmiepatel720]

@@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?

[@koshka02]

Why can't my professor teach like this , this is so easy thank you for making this so straight forward

[@WrathofMath]

So glad it helped, thanks for watching! A few related lessons if you're interested...
Counting Partitions and Bell Numbers: ua-cam.com/video/iJF2kPFGTUo/v-deo.html
Recurrence Relation for Bell Numbers: ua-cam.com/video/sPGudyLalmE/v-deo.html
Recurrence proof: ua-cam.com/video/abfCpVASfLM/v-deo.html

[@tomcat1112k]

you explanation helped me to understand partitions more clearly than a formal book. thank you for making videos like this

[@WrathofMath]

So glad to help!

[@vishnum3690]

Thank You! Can you please do a video on the recurrence formulation of the Bell number(number of ways to partition a set)?

[@WrathofMath]

Thanks for watching, Vishnu! And great idea, I'll get right to work on that lesson! Thanks for the request!

[@vishnum3690]

Thank you so much!

[@remilff7902]

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!

[@yamatanoorochi3149]

This video was way better than the shabby uni provided vid thank you

[@piichuu3596]

Absolutely amazing video. Clear explanation and the presentation is 100/10

[@WrathofMath]

Thank you!

[@randyrandall1767]

So much more clear than other videos.
Thanks.

[@WrathofMath]

I'm glad it was clear, thanks a lot for watching! Let me know if you ever have any questions!

[@MaxPicAxe]

Definitely one of the best UA-cam channels!

[@WrathofMath]

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: ua-cam.com/video/iJF2kPFGTUo/v-deo.html
Recurrence Relation for Bell Numbers: ua-cam.com/video/sPGudyLalmE/v-deo.html
Recurrence proof: ua-cam.com/video/abfCpVASfLM/v-deo.html

[@robertdowneyjr.3558]

You made this look so simple and kinda cool. Thanks from India!

[@WrathofMath]

Thanks for watching, so glad it helped!

[@pavanajsridhar939]

damn!! the best explanation of partition I have ever encountered!! It was clear and to the point!

[@sparrowp2251]

this is just perfect ,loved it thank you

[@ayeshaumairjameel4041]

Excellent explanation . . easy to understand. . . . . Thank u very much😇

[@WrathofMath]

Glad to help, thanks for watching!

[@thestar001Official]

Smooth lines.. smooth teaching 👌

[@WrathofMath]

Thank you!

[@pacsrighthand5923]

Thanks this is really helpful you're very good

[@WrathofMath]

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)

[@sabinomuniz5542]

Explained it so simply. Much thanks!!

[@WrathofMath]

My pleasure, thanks for watching! A few related lessons if you're interested...
Counting Partitions and Bell Numbers: ua-cam.com/video/iJF2kPFGTUo/v-deo.html
Recurrence Relation for Bell Numbers: ua-cam.com/video/sPGudyLalmE/v-deo.html
Recurrence proof: ua-cam.com/video/abfCpVASfLM/v-deo.html

[@karolina6775]

Thank you for the explanation

[@WrathofMath]

No problem, thanks for watching!

[@banelenyide9087]

clear and straight to the point

[@WrathofMath]

Glad it was clear, thanks for watching!

[@SoySauseManGames]

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

[@kabangukabangu2529]

The visual made it clearer 👏🏿

[@WrathofMath]

Glad to hear it, thanks for watching!

[@arnavchandrayan2349]

Nicely described in short time

[@WrathofMath]

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!

[@leocheng9428]

clear and clean af

[@WrathofMath]

Thanks for watching!

[@luka1696]

Very nice explanation. Made it easy.

[@cosmicmelody4259]

Thank you soo much,, your way of explanation is soo smooth ❤❤

[@alexandriagladys1201]

you're the best! this made it so easy to understand

[@WrathofMath]

So glad it helped! Thanks for watching and let me know if you ever have any questions!

[@WOLF91]

Appreciate this. Do you have a video on partial orders and composition?

[@MosaAhmad-sz4yt]

Fruitful explaining

[@WrathofMath]

Glad it helped!

[@swayenkumar2776]

thank you.. great lecture... I am from bangladesh

[@shailendrarajput6053]

Great way of explanation.🇮🇳

[@WrathofMath]

Thank you! So glad it was clear!

[@SaGaR-is1jg]

short and straight to the poitn w for you

[@sristysingh001]

Great explanation
Just cleared my confusion

[@WrathofMath]

Thank you, glad to help!

[@ayang315]

Excellent explanation! Thank you.

[@WrathofMath]

Glad to help!

[@yashodharmasingh3889]

Thank you sir

[@WrathofMath]

My pleasure!

[@rohitritthetend5501]

Best explanation.....✨✨🤗

[@zazatjijombo174]

Straight to the point thank you sir ❤

[@WrathofMath]

Glad to help!

[@hrishikesh_00]

Is there any formula to calculate how many partition can be done for any number

[@con_el_maestro3544]

Thank you very much sir

[@sourabhkaushik7687]

Best explained thank you so much for your effort

[@WrathofMath]

Glad it helped! Check out some of my related lessons for more on partitions...
Counting Partitions and Bell Numbers: ua-cam.com/video/iJF2kPFGTUo/v-deo.html
Recurrence Relation for Bell Numbers: ua-cam.com/video/sPGudyLalmE/v-deo.html
Recurrence proof: ua-cam.com/video/abfCpVASfLM/v-deo.html

[@dianeyeoman8046]

Clear and concise!

[@WrathofMath]

Thank you!

[@firasdjebbi6737]

thank you!! you made so easy

[@WrathofMath]

Glad to help!

[@Alter_Singh]

Very good explanation

[@WrathofMath]

Thank you, glad you found it clear!

[@vysakhmusicspace9634]

Thanks a lot sir❤️....
Awesome session...👍

[@WrathofMath]

You're very welcome, thank you for watching and let me know if you ever have any questions!

[@GradientSoln-En]

Thanks

[@WrathofMath]

Welcome!

[@Flux281]

what if the set S has repeating elements , do write it twice in P or do i just follow the definition?

[@mohdfaiz5977]

remember a set doesnt contain repeating elements.

[@user-fg1ix4pj6i]

Thank you!

[@user-tp1pi3um1m]

Nice explanation.thank you,sir.👍🏻

[@WrathofMath]

My pleasure, thanks for watching!

[@_Maha1]

This is usefull!!! Thank u so much

[@WrathofMath]

You're welcome!

[@swagatikadas6664]

Very useful....🙂
Thank you very much...😃😄

[@WrathofMath]

You're very welcome, thanks for watching!

[@believerslightministriestv573]

Is there any formula that helps us to know the number of partition a set should have? If yes what is it?

[@WrathofMath]

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: ua-cam.com/video/iJF2kPFGTUo/v-deo.html
Recurrence Relation for Bell Numbers: ua-cam.com/video/sPGudyLalmE/v-deo.html
Recurrence proof: ua-cam.com/video/abfCpVASfLM/v-deo.html

[@ski34able]

Amazing!

[@WrathofMath]

Thank you!

[@christydevaraj8411]

Any easy formula to find partition of large numbers

[@winonaeliseragabautista8021]

can i put six numbers in a subset? (like {{1,2,3,4,5,6}}
thank you.

[@akujobi620]

This was perfect, thanks

[@WrathofMath]

My pleasure, glad it helped and thanks for watching!

[@debabandanpattnaik3274]

Nice explanation

[@WrathofMath]

Thanks Debabandan!

[@miss_r801]

Thankyou sir💜

[@ahsanulhaque4811]

Immediately subscribed after seeing the channel Name. Such a cool Name :D

[@WrathofMath]

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: ua-cam.com/video/29qzzNEmEOc/v-deo.html

[@taljune142010]

Great! Thanks!

[@WrathofMath]

Glad to help - thanks for watching!

[@yamatanoorochi3149]

What cover of moon river is in your outro?

[@Victual88]

Thanks!

[@WrathofMath]

No problem!

[@taotechsolutions]

Yes
Thanks

[@arjunarvind2178]

well explained sir

[@WrathofMath]

Thank you!

[@spacecommunityv.i.d.a2040]

Love this video keep it up 👍

[@WrathofMath]

Thank you very much!

[@creative_galaxy490]

plz upload the video for cross partition

[@moon3252]

Thank u so much

[@WrathofMath]

Most welcome 😊

[@dariazhivaliukova819]

studying in Germany, being russian, watching math explanation in english
thanks

[@WrathofMath]

That's incredible! Thanks for watching, so glad to help!

[@itsev6970]

Excellent

[@WrathofMath]

Thank you!

[@ahmetkarakartal9563]

THANK YOU SOOOO MUCH

[@WrathofMath]

You're very welcome, thanks for watching!

[@ahmetkarakartal9563]

@@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.

[@WrathofMath]

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.

[@ahmetkarakartal9563]

@@WrathofMath Okay, thanks again for helping to me

[@AabhusanAryalOfficial]

Thanks :)

[@WrathofMath]

You're welcome! :) Thanks for watching!

[@sandeepsahanicodes]

THANK YOU

[@WrathofMath]

You're welcome! Thank you for watching!

[@Yzyle]

what if (1,3) (2), (1,3), is it still partition?

[@time_lightsout_1186]

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.

[@apeximperator]

TYSM😁

[@WrathofMath]

You're welcome! Thank you for watching! I like how your emoji matches Naruto's face in your profile picture!

[@apeximperator]

@@WrathofMath woahhhh, then you're my sensei 😃

[@WrathofMath]

Well, frogs and toads happen to be my favorite animals! I’ll be the Math Toad Sage!

[@apeximperator]

@@WrathofMath ohhhhh, Jiraiya Sensei😎

[@aqsaameer3403]

Great sir

[@WrathofMath]

Thank you!

[@tejaswinibonam3170]

In How many ways can we do partition of a set?
Plz..reply sir

[@tejaswinibonam3170]

Plz reply sir🙏🏻

[@danielwolff515]

Depends on how big the set is

[@finethehuman1821]

Only as many ways as the number of elements in the set you want to partition

[@con_el_maestro3544]

Love how you gave up on drawing curly brackets correctly 😂

[@proviptk]

2:06 I thought the third one contains 3...

[@alishanokomochian8302]

please come teach in our university please

[@jeremyvernon8748]

Noooowwwww I get it. Thank you!

[@WrathofMath]

Glad it helped and thanks for watching!

[@godfather-gh2vq]

Thanks😊

[@WrathofMath]

You're very welcome! 😊 Thanks for watching!

[@nam1820]

What is the formula for finding partition of a set

[@WrathofMath]

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).
ua-cam.com/video/iJF2kPFGTUo/v-deo.html
ua-cam.com/video/sPGudyLalmE/v-deo.html
ua-cam.com/video/abfCpVASfLM/v-deo.html

[@a.nelprober4971]

Youre a beast

[@WrathofMath]

I do my best - thanks for watching!

[@tilugulilwa]

Awesome

[@WrathofMath]

Thank you!

[@ryoukboy1010]

6:09. Green elements...🤨😦

[@VoidTailor]

nice

[@shahin4959]

Sir please hindi and English mix karke is questions Ko batai n

[@nabman5619]

good

[@WrathofMath]

Thank you!

[@musicstation679]

who made it better then my lector