How to do a PROOF in SET THEORY - Discrete Mathematics

We learn how to do formal proofs in set theory using intersections, unions, complements, and differences.
0:00 - [Intro]
0:49 - [Language of Set Theory]
3:31 - [Proof #1]
6:15 - [Proof #2]
11:12 - [Proof #3]
14:25 - [Proof #4]
#SetTheory #Proofs #DiscreteMath
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Discrete Mathematics 1: • Discrete Math (Sets, L...
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Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.
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- Nikita Tsyganov

[@Trevtutor]

Check out my new course in Set Theory: trevtutor.com/p/master-discrete-mathematics-set-theory It comes with video lectures, text lectures, practice problems, solutions, and a practice final exam!

[@Trevtutor]

How many of you wanted to cry the first time you had to do one of these? :'(

[@pawelbialkowski8296]

I've just been given a task to make a proof in set theory (cardinality) and I had no idea how to begin. I understand the formula and it makes logical sense but didn't know how to prove it. Thanks for this video.

[@sarahtipper559]

yes sirrrr

[@melaninshotz_6]

Please can you help me with this :( ...... (AUB)-(C-A)=AU(B-C)

[@fahimahadam7026]

definitely me, still trying to wrap my head around how to do it.. but this video helped me to understand it better

[@sainthentai7763]

Where dom ? Where max and min sets ? Pls speak me your location this themes.

[@yuto2497]

I just realized that both the intersection (Upside-down U) and the union (U) symbols also correspond to the logical operator AND (Upside-down V) and OR (V).

[@Carrymejane]

Yes actually before u learn this u better learn logic first

[@dankdreamz]

Thank you! You managed to reignite my passion for the subject matter that my professor managed to extinguish.

[@lebraunJaymss]

dude i'm crying ahhahaha

[@spicy5845]

Thank you, you have a gift of explaining it well and understandably for beginners!

[@shawntyler7798]

literally uploading along with my semester's schedule, thank you

[@daisydelaneyful]

You are a lifesaver! I have a huge test on monday and I can't stand set theory, so grateful for this.

[@Deksudo]

Thank you for blessing us newcomers with freshly uploaded content in the middle of a half-decade-old playlist

[@megamehdi89]

I wish you could upload more often, I love watching and learning from your videos.

[@Trevtutor]

More to come!

[@megamehdi89]

@@Trevtutor That will be fabulous. Thank you for your effort and time.

[@user-hc4ci1yq1z]

thank you so much for making these videos! i feel more confident of doing set proofs now

[@emiliasuero310]

I found it so difficult to underestand how to proof, and with this video is it now more clear to me, THANK YOU

[@DJSaez-ll8or]

Excellent video, have a test in two days and this was exactly what I needed! Thanks a lot

[@luiza3286]

Thank u soooo much for this video. U saved me so much! Shared the channel with all my classmates so that we can all ace our upcoming midterm!

[@sayeddileri3461]

I'm already learning so fast!!! Thank You.

[@benjaminlens2384]

Thank you for your time and effort. The first time that math excites me again.

[@thebeeskhakis7145]

just started grad school and this was what i needed. thank you!!

[@aaronroberts7119]

Wonderful, clear, THANK YOU!

[@absence9443]

Neat introduction, I do feel more confident with the basic proofs, however once encountering functions everything starts feeling less intuitive or formulaic but rather abstract especially since you cant visualize it anymore.

[@miguelavalos1316]

i needed this you saved my life!

[@drakefist5693]

Thank you so much! It’s so much clearer now

[@sagejpc1175]

He's BACK!

[@aidan7294]

This video is brilliant and has been a massive help in making me understand set theory! Thank you so much!

[@cambridgebreaths3581]

Yay. Back again. Thank you

[@andristic]

Nice simple tutorial. I'm using free texts and they are great but it's so handy to have someone walk me through a couple proofs.

[@hidden6955]

Thanks alot that really connected the pieces in my head

[@krissy_n5732]

Amazing video mehn. Saved my life!!!

[@Harsh-un2zo]

Thank you sir now we are able to do each and every question ☺️

[@witness9147]

Thanks, your videos helped me a lot

[@ethanbrooks9832]

thank you brother keep doing good work

[@Katinka6413]

Much appreciated! 🤓

[@asusvivobook3566]

More videos for proofs like this plz !!!

[@apaubright5540]

in fact, this man is good

[@andrewdupree8947]

Great video to refresh my memory for an upcoming exam. I'm not too sure if the graders at my university would be okay with 7:24 though. I think 'x (is not an element of) B n C ' would have to be rewritten as 'x (is an element of) the complement of B n C' (by definition of complement) and then you'd have to use De Morgans for sets. Or if the graders are super picky you'd have to prove using logical equivalences and set builder notation.

[@anthonytran1912]

Nice your back again

[@suspatrol9550]

Thank you so much, lowkey you have saved me on my course

[@Trevtutor]

I’m glad I could help!

[@xx_foxpvp_xx374]

sir you are just amazing

[@MuninnNova]

You just saved my homework, thank you! I wish you were my teacher T.T

[@shivanshnigam4015]

Thanks so much I have an exam tomorrow and I didn't understood when the teacher taught it

[@adammarrouny7772]

Thank uuu sooo much sir

[@memberre660]

What tools are you using in your tutz like this one?

[@bonganipeter9433]

Thank You.

[@JMRG2992]

This was more educative than my master's course in mathematics. Thank you.

[@oluwaseyialaya7578]

Thank you sir

[@leovoghera6174]

Thank you for explaining the topic that made me wanna smash my head against a wall in such a comprehensive way

[@pedro2908]

Hi. Loved the video. One question: Did you use De Morgan's laws in 7:13?

[@martinjoroge]

yeah he did, plus your comment saved me a lot of thinking

[@aliawde4970]

Nice video ❤🔥

[@mohamedgames783]

you are the best

[@PJ-nc4jh]

Hi, i tried to get access to your workbook via the bitly link you posted however I am unable to get access to it due to privacy reasons

[@euanliew967]

Hi,for 12:18 where u wrote x does not belong to AUB, why did you write 'and' instead of 'or' for the next line? I thought U operators used 'or' instead? Thank you.. Edit: Does it work oppositely when you apple e with a slash?

[@Trevtutor]

The complement changes things. If I say “I am not a man or a whale” then you can infer that “I am not a man” and “I am not a whale”. This is the same idea.

[@JugalSingh]

@@Trevtutor I'm still very confused on where to write "and" and where to write "or". You said we use 'or' for union and 'and' or intersection. There are points where you did the opposite ( I thought that was because of the complement or because of the c with the /) but at some instances you didn't even follow that pattern.

[@Deksudo]

​@@JugalSingh I had the same confusion and thinking of the expressions visually helped me. "AuB", "A or B" includes everything in both A and B, the entire are of those two sets be colored if you draw a representation of it. So when you have an expression like "x !€ (AuB)", which means x is in neither of those sets, and you want to expand that, you have to do it without changing it's meaning. If you expand it as "(x !€ A) OR (x !€ B)" fulfilling just one of those two conditions would satisfy the requirement, so an element from the set B would be in that set because it isn't in the set A. But your original expression requires that it won't be in any of those sets A or B, so you should write "(x !€ A) AND (x !€ B)". Try drawing them on a paper. Alternatively, when reading the expression "(x !€ A) OR (x !€ B)", imagine a bucket you fill with elements from both sets as you read the expression. The part inside the first parentheses says all x that are not an element of A, so you say okay and add the things inside of B, because they're not in set A after all. You get to the second part, the second brackets, and it asks for all x that are not elements of B, and you say okay again and add everything from A. So you didn't want anything from A or B but you have everything in them except their intersection, their intersection is the only place in the universe that doesn't fulfill either of those conditions.

[@sharonlu7347]

my math class makes more sense now 🥲thank you

[@nosferatee]

Super!!!🔥🔥🔥

[@aaraizm]

Thank you so much. Better than my prof

[@arnoldmukucha3359]

Hello. On the question A'nB' C AUB complement. How come u got x E AUB complement, is it not supposed to be x E AnB complement, since and is for the intersection

[@HopeCoolson]

That video and chanel is a gift from god

[@Trevtutor]

I appreciate it :)

[@christianhill3244]

idk i got confused but on the first proof i got x is an element of a such that for all in a(compliment) equals b(compliment)....or approaches b

[@user-dz2wb6hi8w]

At 14:00, how come on the left side for 1, 'and' turns into ^, (from step 4 to 5); but on the right side for 2, 'and' turns into U (from step 3 to 4)? (I don't have the proper notations but they're similar)

[@nithin.j7910]

can list of anything be called set or there are some rules

[@aziza7806]

at 7:50 why OR is written for Intersection?

[@krran792]

Trevtutor better than any online edu company!!!

[@krzysztofjarek6476]

Thanks for your work 🙂 In 12:47 in the third line of proof (1) there shouldn't be 'or' instead of 'and'?

[@krzysztofjarek6476]

And similarly for the third line in proof (2) (e.g. 14:00)?

[@1eeee806]

In min 12:10 how A Union B becomes (AND)in the next step, i think union should be (OR)

[@liyangxx]

yeah i need an explanation too

[@passavudhsabpisal9741]

how????

[@hydrarl3869]

Union is "or" but the negation of the initial statement makes the union an intersection

[@scotwot1]

why is x not in B compliment C? or would this still be correct?

[@jamieha2035]

God damn where have you been all my life?

[@nasertizhoosh9434]

At minute 12:50 you put X is not a member of A AND X is not a member of B ( I think I got confused there since shouldn't Union indicates or)

[@furoebiere6756]

Please analyse the following question cos i'm a bit confused • A={x: x2 = 9, x 3 = 5 } (11) B {x: x2 = 9, x 3 = 5}. thank you

[@karma_akabane1304]

for the last question if you say but,A is a subset of b then x is a element of b ,if x is a element of b then x is not a element of c therefore x is and element of B not C is it still correct

[@danilojonic7924]

You subconciously applied one of the De Morgan's laws in Proof #2 with B^c and C^c being equal to -B and - C accordingly. So from x not in (B and C) we got to x not in B or x not in C if this transition from 'and' to 'or' confused some of you here's a good explanation of it. Written with logic grammar: -(B and C) (-B) or (-C)

[@hamzaelmi5584]

If you used B - C as your assumption in your last example, would proving the 'if', 'then' statement still be possible

[@eugeneassoahmorcher2042]

Can you please make a video and prove : A intersection Bcompliment all in one bracket UNION in another bracket Acomplement Intersection B .

[@harshkapadia5824]

Can we prove using examples?

[@willm3889]

At 7:57 why did you put "or" between the xeB' "or" xec'? The original statement is xe' BnC where "n" is the intersection and by def of set means "and" not "or". So why didn't you write xeB' "and" xec'???? How would I know to put the "or", when the "n" symbol means "and"???

[@Trevtutor]

Not (happy and sad) is equivalent to (not happy) OR (not sad) if we think about it in plain English

[@willm3889]

@@Trevtutor Thank you

[@gackerman99]

"it's not that hard" except when I tried the last proof I started with "assume x is in A, therefore X is in b" which doesn't get you anywhere. so there's a critical detail missing here of where you're supposed to take your generic sample from.

[@borgin1031]

Why was x not an element of A union B considered same as x not in A and x not in B? Union means or right?

[@Trevtutor]

If I say I’m not (a cat or a dog), does it imply that I can still be a cat? It implies both not a cat *and* not a dog.

[@user-ls9hj4wm3i]

❤❤❤

[@NengiWillie07]

i was taught that AUB is the same thing as everything in A and B. But you're saying A or B could you shed more light on that

[@Trevtutor]

Yours is another way of saying the same thing, but the logical word we use in math for that situation is “or” when it comes to the union.

[@NengiWillie07]

@@Trevtutor thanks a lot clears up a lot of things for me

[@mattiavitodangelo411]

why not use truth tables?

[@Carrymejane]

It's confusing since he's teach set first then logic, rather than truth tables u better understanding the laws of set table it has the same idea as algebraic precedence

[@notalexj]

fucking legend

[@gackerman99]

what if A is the empty set? are we still allowed to talk about "some generic element of A"?

[@vincenzocassano5609]

(A-B) U (A n B) = A'

[@_7__716]

I dont really follow how this proves the two are equal though. 14:17

[@raheem2845]

I would claim sets are built from images. But first I will show that numbers are built from images Example , 4 always represents 4 images, like 4 squares for instance. To be specific numbers are "labels" for groups of images 1. The main idea here is that maths is built from images (a) example , geometry is clearly made of images b) example 2, We claim numbers are built from images too, as say 4 , always represents 4 images, like 4 squares for instance. C) imaginary numbers are connected to images too , which is why they have applications in physics D) In general any mathematical symbol that comes to mind is connected to images too.

[@flann3l102]

For proof #2, why does x HAVE to be an element of b compliment or c compliment. For example, if we suppose x is an element of A - (B and C) isnt it entirely possible that x is only an element of A? If this was the case, the suppose statement would still be true and the proof method you use would be incorrect to claim that x is an element of b compliment or c compliment. I understand the proof but I'm hung up on exactly why we can claim that x is an element here when it technically doesn't have to be?

[@Mercedes-Scott]

Thank you, you have a gift of explaining it well and understandably for beginners!

[@user-oq5un6cs7v]

Bless up for your good work. You indeed know how to explain these stuff