Cardinality of the Power Set Part 2
The cardinality of the power set is never the same as the cardinality of the original set. This can be proven with Cantor’s diagonal argument familiar from the proof of the uncountability of the real numbers.
The cardinality of the power set is never the same as the cardinality of the original set. This can be proven with Cantor’s diagonal argument familiar from the proof of the uncountability of the real numbers.
Пікірлер: 17
This proof is very similar to Cantor’s diagonal argument. Very clear presentation. You should do a video on friendly numbers in number theory.
beautifull, your explanation was superb as always.
Thank you very much, these videos are truly well made!
Amazing videos mate, keep it up, very bingable!
Your math videos are a joy to watch
Thank you so much ..... I am in love with your videos your explanation and drawings make everything easy ..... No enough words to thank you..... Plz could you make videos on cancer stem cells .... idiopathic pulmonary fibrosis...flow cytometry.... Thanks again and good luck
These videos are excellent! Will you ever make more videos on set theory?
Would you please upload video on Solid phase Oligodeoxynucleotied synthesis.(Triester pathway and Phosphoramidite pathway)
Omg you saved my life !!! I love you
@elliotnicholson5117
3 жыл бұрын
You’re welcome.
I really hope you do a series on quantum mechanics someday.
Thanks a lot for all your great videos! The argument is truly beautiful but its similarity with Cantor's diagonal argument makes me wonder: is the proof valid for all infinite sets or only for countable infinite sets?
@elliotnicholson5117
5 жыл бұрын
In principle there still exists a set outside of the range of any attempt at a bijective map even if the set is uncountably infinite.
Why does my textbook says that it is enough to show that there is an injection and not a surjection to show that the cardinality of any set A is less than the cardinality of power set A
@MuffinsAPlenty
Жыл бұрын
That actually is enough. There is a subtle difference here. What Elliot is saying is that if you are dealing with finite sets S and T and you find a single function f : S → T which is an injection but not a surjection, that is enough to conclude that |S| However, with infinite sets X and Y, it is possible to have a function f : X → Y which is an injection but not a surjection while simultaneously having another function g : X → Y which is a bijection. What your book is saying is that, in order to show |A| And this is the difference between "there is an injection _which is_ not a surjection" (enough for finite sets) and "there is an injection _and_ not a surjection" (needed for infinite sets).
Which software you use to teach.??
@elliotnicholson5117
3 жыл бұрын
Explain everything.