countable union of countable sets

The Union and Intersection of Two Countable Sets is Countable @benssiao Are you sure the textbook doesn't say "countably infinite"?Isnt countable and countably infinite the same? . Countable may also be finite. I have Stephen Abbott's second edition of Understanding Analysis. Watch headings for an "edit" link when available. If you want to discuss contents of this page - this is the easiest way to do it. However my textbook says otherwise. Countable may also be finite. I am looking at theorem 1.5.8 ii on page 29. Now A∪B={cn:n∈N} and since it is a infinite set then it is countable. How come? In the sense they are the same as saying a function is a bijection between N and A where A is any set. However my textbook says otherwise.

Also it says that numbers arranged into a square like soproves this theorem. I then began to think we can use induction to say that the countable union of countable sets are also countable.

1 3 6 10 15 . Proof. My textbook must have a type because it claims that an infinite union of countable sets is countable. 7 12 . Thanks alot. AC ω is particularly useful for the development of analysis, where many results depend on having a choice function for a countable collection of sets of real numbers. However my textbook says otherwise. ZF + AC ω suffices to prove that the union of countably many countable sets is countable. @benssiao What book is that, and what page is the typo on, so we can all correct it in our copies? @benssiao What book is that, and what page is the typo on, so we can all correct it in our copies? .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; Also it says that numbers arranged into a square like soproves this theorem. . Thanks alot. @benssiao Not entirely. I am looking at theorem 1.5.8 ii on page 29.Isnt countable and countably infinite the same? I have Stephen Abbott's second edition of Understanding Analysis. In the sense they are the same as saying a function is a bijection between N and A where A is any set. But even if you mean countable as "countably infinite", still, the terms Awesome! I fail to make the connection.I understand how to prove that the union of 2 countable sets is countable. For example, take the setsHowever, your "square proof" would work fine to prove the statementThe proof would work because you can map the set $mathbb N$ to the countable union of countable sets, by mapping $1$ to the first element of the first set, then $2$ to the first element of the second set, $3$ to the first element of the first set, $4$ to the first element of the third set, and so on.This statement would be very hard to prove by induction, however, because induction can only ever prove statements that are true An infinite union of countable sets may not be countable. \begin{align} \quad h(c) = \left\{\begin{matrix} 2f(c) & \mathrm{if} \: c \in A \\ 2g(c) + 1 & \mathrm{if} \: c \in B \end{matrix}\right. Then we can define the sequence (cn)∞n=0 by c2k=ak and c2k+1=bk. @benssiao Not entirely. The union of two countable sets is countable. Let A={an: n∈N} and B={bn: n∈N}. Countable may also be finite. In the sense they are the same as saying a function is a bijection between N and A where A is any set. I fail to make the connection.I understand how to prove that the union of 2 countable sets is countable. Thanks alot. @benssiao Are you sure the textbook doesn't say "countably infinite"? I have Stephen Abbott's second edition of Understanding Analysis. I have Stephen Abbott's second edition of Understanding Analysis. Find out what you can do. For example, take the setsHowever, your "square proof" would work fine to prove the statementThe proof would work because you can map the set $mathbb N$ to the countable union of countable sets, by mapping $1$ to the first element of the first set, then $2$ to the first element of the second set, $3$ to the first element of the first set, $4$ to the first element of the third set, and so on.This statement would be very hard to prove by induction, however, because induction can only ever prove statements that are true Thanks for contributing an answer to Mathematics Stack Exchange!Use MathJax to format equations. But even if you mean countable as "countably infinite", still, the terms Awesome! In terms of cardinal numbers and their arithmetic the cardinality of a countably infinite set is aleph-null , and … Countable may also be finite. \end{align} With the foresight of knowing that there are uncountable sets, we can wonder whether or not this last result can be pushed any further.

For example, take the setsHowever, your "square proof" would work fine to prove the statementThe proof would work because you can map the set $mathbb N$ to the countable union of countable sets, by mapping $1$ to the first element of the first set, then $2$ to the first element of the second set, $3$ to the first element of the first set, $4$ to the first element of the third set, and so on.This statement would be very hard to prove by induction, however, because induction can only ever prove statements that are true An infinite union of countable sets may not be countable. A few useful tools to manage this Site.

.... proves this theorem.

4 8 13 . My textbook must have a type because it claims that an infinite union of countable sets is countable.Awesome!

My textbook must have a type because it claims that an infinite union of countable sets is countable. All relevant content documents on this site are for trial only Please support original, if someone is involved in legal issues This site does not bear any consequences Also it says that numbers arranged into a square like so. View/set parent page (used for creating breadcrumbs and structured layout). I have Stephen Abbott's second edition of Understanding Analysis.

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