Countable and Uncountable Sets - Discrete Mathematics
ฝัง
- เผยแพร่เมื่อ 18 ธ.ค. 2022
- In this video we talk about countable and uncountable sets. We show that all even numbers and all fractions of squares are countable, then we show that all real numbers between 0 and 1 are uncountable.
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I just realized this video in the DM1 series was recently posted. A massive thanks to you for continuing to update these playlists and get this guy in his 30s off to a great start in DM1 for a CS degree!!
I think this video is a bit overdue by now. But here we go: countable and uncountable sets!
I started learning this 3 days ago, im so glad you released a video on it! whenever I have problems understanding a discrete mathematics topic I instatly google the keyword and your name xD Thank you so much for content!
I literally have an exam in 3 hours, and I couldn't find a video on countable sets from you last night but here you go, saving my life last minute
The timing is impeccable ;)
count... yourself 🍀
How was the fail?
@@shayorshayorshayorgot an A
Sameee broo😂
Thank you so much dude! Aside from the helpful content, the pacing of this video is perfect.
I've watched soo many other math, coding, etc. videos where they rush through the content or speak too quickly and it makes an otherwise informative video incredibly frustrating and worth disliking.
Glad I found your channel!
Wouldn't the uncountablity proof work for the natural numbers, too?
thank you sir, very well explained.
afaik in this playlist we haven't yet talked about bijections/surjections etc. so this was a bit abrupt.
my exams is tomorrow thanks god, helped me
Can you help me with this?
11. (15 points) Draw an undirected graph with six nodes and nine edges. Label the nodes 1
through 6. Write down the formal 2-tuple describing your graph. What is the
lexicographically first maximal independent set of your graph? Is it a maximum independent
set? Explain why or provide a maximum independent set.
Will that function di always work?
Thank you ❤❤❤
R/Q = Irrationals. All point in this set are an acummlation point?
Amazing video
2:18 why are you able to exclude zero?
Not sure the playlist is in order
Yep i do think the playlist isn't in order 😌
amazing
So is it not a contradiction that we generated a real number greater than 0, but less than 1 and claimed for it not to be in the set of real numbers less than 1 and greater than zero? Surely this should reflect that the method breaks down somewhere?
That contradiction is what enables us to say it’s uncountable. If it were countable we wouldn’t be able to get to a contradiction.
@@Trevtutor oh right, that makes sense.
Thank for the video and the proof, but I am a bit confused. Wouldnt this make the natural numbers uncountable as well?
Lets say I have
a1=1
a2=2
a3=3
Now I go through all and keep appending them.
so a4=123
When I get to a123, the new number would get 123 appended at the end and hence wouldnt be in the set.
What am I missing?
The proof for rationals differs from integers as for every two rational numbers x and y, you can always create a new number (x+y)/2. With integers, there are no numbers between, so we can order them according to the video without asking “what about the numbers in-between?
For example, with rational numbers, what’s after 0.0001? 0.00001? 0.0002? Well, if 0.0002 is next, then I can create 0.00015 between the two. Then I can make a number between 0.00015 and 0.0002. Etc. there’s always a new number available to create. The proof of rationals and it’s contradiction gets at that idea.
@@Trevtutor thanks for the reply. You gave the example of rationals. But aren't the rationals countable?
I understand how to make the new number but I don't understand what is its purpose. To prove that although we make a new set from original set, the new set is still uncountable?
Assume there is a set of real numbers between 0 and 1 that is countable, which means there is a 1-to-1 mapping between the set and the natural numbers. Now we can create a new number, which is still a real number between 0 and 1, add this new number to the previous set to make a new set, so the new set does not have a 1-to-1 mapping with natural numbers. This new set is still a subset of all the real numbers between 0 and 1. So all the real numbers between 0 and 1 does not have a 1-to-1 mapping with natural numbers, so it is uncountable.
@Kai Tan I get it:3 Thank you 😊
@@kaitan8824 great explanation. I had no idea how to do this in my discrete structure lecture. I get how to do this after watching the video but don't know why. Your explanation clearly explained everything!
I feel very clever now 😮
Hey i didn't get the a¹¹ part
2:10 isn't it because 0 is not in the natural number set ?
2:12
Sorry
Is zero in the set of natural numbers?