This is the first in a series of tutorial videos that I am making, aimed at undergraduate maths students at Cardiff University (and anyone else who wants to view them). The videos will cover various topics taught at undergraduate level. *Countable and Uncountable Sets:* In maths, a set containing infinitely many objects may nevertheless be countable. This video explains how to determine whether or not an infinite set is countable, and what this means in a practical sense. It also examines the countability (or uncountability) of some standard sets of numbers, including the integers, rational numbers and real numbers. Part 1: Countable and Uncountable Sets (Part 1 of 2) Part 2: Countable and Uncountable Sets (Part 2 of 2)
Absolutely brilliant explanation. I had been so confused on what cardinality and countable sets meant but you just explained those topics effortlessly and in such clarity!
Thank you, older student this is one of my last two classes for my computer science degree and I felt like I was getting the bums rush on this material. You have helped a lot on this...simplify then depth. Can't seem to get that understood on this end.
Great explanation. It helped me understand countable infinite sets which is crucial for me to answer ' Prove that the set of all two dimensional Cartesian coordinates with integer coordinates form a countably infinite set.'
Excellent, excellent, excellent presentation! Trying to unravel this subject, I now see that I have a different definition of "countable". It appears that you are going to talk more about this in Part 2. I do feel a glimmer of understanding seeping in. Just because something is "countable" does not mean it can or will ever be counted.
Could you please explain the mistake in Mueckenheim's argument? If all positive fractions can be enumerated, then the natural numbers of the first column of the matrix 1/1, 1/2, 1/3, 1/4, ... 2/1, 2/2, 2/3, 2/4, ... 3/1, 3/2, 3/3, 3/4, ... 4/1, 4/2, 4/3, 4/4, ... 5/1, 5/2, 5/3, 5/4, ... ... can be shuffled such that they cover the whole matrix. But by exchanging them with other fractions, never the whole matrix will be covered.
So when we take integral we say in Riemann sum that ve define n parts and say n goes to infinity is the integral but İntegral talks about uncountable Numbers besides Riemann sum talks about countable objects, so is this a contradiction ?
Im having a thought time comparing languages of 2 or 3 things: cardinality, countable and bijectivity. If there is a bijection between 2 groups that means they have the same cardinality? The same size? |A|=|B| ???? My question arises because of the fact I keep reading definitions that say: we can say a group is countable IF it has a bijection with the Natural numbers, but that seems to be wrong, because IF the group is finite it has a bijection with a Finite set of the Natural numbers, If the group is infite it has a bijection with the Natural numbers. Is that correct? Am I understanding the definition incorrectly?
I know I am many, many years late, but I'll explain two issues that are coming up here. 1. There's a difference between *_the_* set of natural numbers and *_a_* set of natural numbers. a. *_The_* set of natural numbers is the full set of _all_ natural numbers. b. *_A_* set of natural numbers is any subset of the natural numbers (a set where every element is a natural number), which could include finite sets. So the article used (a or the) matters. 2. Some people use different definitions of countable, actually! a. Some people define "countable" to mean there is a bijection with *_the_* (full) set of natural numbers. Under this meaning of "countable," every countable set is infinite. b. Others define "countable" to mean there is a bijection with *_a_* set of natural numbers. So, in this case, "countable" means finite or countably infinite. So you have to be careful of which definition any given textbook uses, because not all textbooks are consistent.
Thank you for posting this, you have down a great job explaining this concept. My opinion is critical of this subject and not your excellent presentation. A set cannot contain an infinite number of items. Physically in the real world this is impossible. In the real world infinity can only be indicated with the idea that we can keep counting forever, adding an extra zero and going from 1's to 10's to 100's to 1000's and so on... In the real world we cannot have a box containing an infinite number of objects. We can imagine an infinite number of fractal elements, but this is just fantasy. We can imagine dividing a line into ever (and infinite) smaller segments. But we cannot actually do this in the real world and this shows us the difference between abstract ideas and concrete ones. A set would indicate some kind of physical correspondence if we are modeling something real. If we are not, then we are just confusing things needlessly by showing the limits of a tool, much in the same manner we would not use a screwdriver when we need a saw. Math is meant to be a tool and it is meant to model reality. It is not meant to model itself. When our forms of communication, like language and math, which are used to model reality, are used to model abstract imagined concepts instead, we run into trouble and end up with 'paradoxes'. But these paradoxes are simply mistakes. This is what happens when we forget that math is a tool. We start to think the tool is the reality and we confuse the two.
I hope you are aware of the fact that natural numbers start with 0, not 1. A set that is either finite or has the same cardinality as the set of positive integers is called countable.
There is no consensus on whether or not the natural numbers start with 0 or 1. He clearly shows that his definition of the natural numbers starts at 1 and is therefore equivalent to the set of positive integers.
This is the first in a series of tutorial videos that I am making, aimed at undergraduate maths students at Cardiff University (and anyone else who wants to view them). The videos will cover various topics taught at undergraduate level.
*Countable and Uncountable Sets:* In maths, a set containing infinitely many objects may nevertheless be countable. This video explains how to determine whether or not an infinite set is countable, and what this means in a practical sense. It also examines the countability (or uncountability) of some standard sets of numbers, including the integers, rational numbers and real numbers.
Part 1: Countable and Uncountable Sets (Part 1 of 2)
Part 2: Countable and Uncountable Sets (Part 2 of 2)
BEST EXPLANATION EVER!!!
i am from kashmir india found ur explanation heart touching clear and convincing.thanks a million sir.
Absolutely brilliant explanation. I had been so confused on what cardinality and countable sets meant but you just explained those topics effortlessly and in such clarity!
Thank you, older student this is one of my last two classes for my computer science degree and I felt like I was getting the bums rush on this material. You have helped a lot on this...simplify then depth. Can't seem to get that understood on this end.
Fantastic!! It is highly appreciated if you could keep posting videos like this.
Extremely clear and useful! Thank you.
Great explanation. It helped me understand countable infinite sets which is crucial for me to answer ' Prove that the set of all two dimensional Cartesian coordinates with integer coordinates form a countably infinite set.'
Excellent, excellent, excellent presentation! Trying to unravel this subject, I now see that I have a different definition of "countable". It appears that you are going to talk more about this in Part 2. I do feel a glimmer of understanding seeping in. Just because something is "countable" does not mean it can or will ever be counted.
Cantor was a genius for discovering this. The diagonal argument was inspired.
can you make a video on ordinal numbers, and ordinal infinity
Superbly explained - thank you.
Very clearly explained-- thanks!
STUDENT: "Math is boring. This is stupid."
TEACHER: "Can you count?"
STUDENT: "Yeah."
TEACHER: "Let me tell you about Georg Cantor."
You have been a great help, thank you.
Could you please explain the mistake in Mueckenheim's argument?
If all positive fractions can be enumerated, then the natural numbers of the first column of the matrix
1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
5/1, 5/2, 5/3, 5/4, ...
...
can be shuffled such that they cover the whole matrix. But by exchanging them with other fractions, never the whole matrix will be covered.
Thanks Rob! Great video mate!
So when we take integral we say in Riemann sum that ve define n parts and say n goes to infinity is the integral but İntegral talks about uncountable Numbers besides Riemann sum talks about countable objects, so is this a contradiction ?
great explanation
Im having a thought time comparing languages of 2 or 3 things: cardinality, countable and bijectivity.
If there is a bijection between 2 groups that means they have the same cardinality? The same size? |A|=|B| ????
My question arises because of the fact I keep reading definitions that say: we can say a group is countable IF it has a bijection with the Natural numbers, but that seems to be wrong, because IF the group is finite it has a bijection with a Finite set of the Natural numbers, If the group is infite it has a bijection with the Natural numbers. Is that correct? Am I understanding the definition incorrectly?
I know I am many, many years late, but I'll explain two issues that are coming up here.
1. There's a difference between *_the_* set of natural numbers and *_a_* set of natural numbers.
a. *_The_* set of natural numbers is the full set of _all_ natural numbers.
b. *_A_* set of natural numbers is any subset of the natural numbers (a set where every element is a natural number), which could include finite sets.
So the article used (a or the) matters.
2. Some people use different definitions of countable, actually!
a. Some people define "countable" to mean there is a bijection with *_the_* (full) set of natural numbers. Under this meaning of "countable," every countable set is infinite.
b. Others define "countable" to mean there is a bijection with *_a_* set of natural numbers. So, in this case, "countable" means finite or countably infinite.
So you have to be careful of which definition any given textbook uses, because not all textbooks are consistent.
Thanks so much! Very helpful.
Amazing job!
great job
Great video!!
Thank you very much for your help!
Very clear, thank you!
I want examples and exercises solved please
great video
Brilliant mate
Thank you sir
good lecture
Fantastic, thanks so much
nice video :)) im studying infinite sets , this is helpful
Great . I m very happy and lucky .
Wow thanks sir
Thank you!
Thank you for posting this, you have down a great job explaining this concept. My opinion is critical of this subject and not your excellent presentation.
A set cannot contain an infinite number of items. Physically in the real world this is impossible. In the real world infinity can only be indicated with the idea that we can keep counting forever, adding an extra zero and going from 1's to 10's to 100's to 1000's and so on...
In the real world we cannot have a box containing an infinite number of objects. We can imagine an infinite number of fractal elements, but this is just fantasy. We can imagine dividing a line into ever (and infinite) smaller segments. But we cannot actually do this in the real world and this shows us the difference between abstract ideas and concrete ones. A set would indicate some kind of physical correspondence if we are modeling something real. If we are not, then we are just confusing things needlessly by showing the limits of a tool, much in the same manner we would not use a screwdriver when we need a saw.
Math is meant to be a tool and it is meant to model reality. It is not meant to model itself. When our forms of communication, like language and math, which are used to model reality, are used to model abstract imagined concepts instead, we run into trouble and end up with 'paradoxes'. But these paradoxes are simply mistakes. This is what happens when we forget that math is a tool. We start to think the tool is the reality and we confuse the two.
finite sets are not one-one correspondance with natural numbers
I hope you are aware of the fact that natural numbers start with 0, not 1. A set that is either finite or has the same cardinality as the set of positive integers is called countable.
There is no consensus on whether or not the natural numbers start with 0 or 1. He clearly shows that his definition of the natural numbers starts at 1 and is therefore equivalent to the set of positive integers.
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