Countability of Sets of Functions and Generalization to Infinite Sets

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In summary, the conversation discusses determining whether a given set is countable or not, using the concept of countable unions and finite products of countable sets. The conversation also mentions a theorem that states the cardinality of a set is equal to the cardinality of its image under a function, and asks for help in finding a bijection between two sets.
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Homework Statement



Determine whether or not the set is countable or not. Justify your answer.

The set Bn of all functions f:{1,2,...,n}[itex]\rightarrow[/itex]N,

where N is the natural numbers.

Homework Equations




1.)A countable union of countable sets is countable

2.)A finite product of countable sets is countable



The Attempt at a Solution



In the solution, a theorem is used that is not in my book.


It goes something like this Cardinality(A)=c and f:A[itex]\rightarrow[/itex]B, then the set of functions is Ba.

I was wondering if anyone could help me figure out what information I was supposed to derive this from?

Thank you.
 
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  • #2
Maybe you can find a bijection between the set of all functions

[tex]\{1,2\}\rightarrow \mathbb{N}[/tex]

and [itex]\mathbb{N}\times \mathbb{N}[/itex]. Generalize.
 

Related to Countability of Sets of Functions and Generalization to Infinite Sets

1. What is the difference between cardinality and countable?

Cardinality refers to the total number of elements in a set or group, while countable refers to a set or group that can be put into a one-to-one correspondence with the natural numbers (1, 2, 3...). In other words, countable sets have a finite or infinite number of elements, while the cardinality of a set can also include uncountable elements.

2. How do you determine if a set is countable?

A set is countable if it can be put into a one-to-one correspondence with the natural numbers. This means that each element in the set can be paired with a unique natural number, and there are no elements left out. For example, the set of positive even numbers is countable because each number can be paired with its corresponding half (2 with 1, 4 with 2, 6 with 3, and so on).

3. Can an infinite set be countable?

Yes, an infinite set can be countable. In fact, there are different levels of infinity, and countable sets are considered to be the smallest level. This means that even though an infinite set may seem larger than a finite set, it can still be put into a one-to-one correspondence with the natural numbers and therefore be considered countable.

4. What is an uncountable set?

An uncountable set is a set that cannot be put into a one-to-one correspondence with the natural numbers. This means that there is no way to count or list all the elements in the set. Examples of uncountable sets include the real numbers and the set of all possible subsets of a given set.

5. How is the concept of cardinality and countable related to Cantor's diagonal argument?

Cantor's diagonal argument is a mathematical proof that shows that there are different levels of infinity. It uses the concept of cardinality and countable to demonstrate that the set of real numbers is uncountable, even though it may seem like it should be countable. This argument helps to explain the difference between countable and uncountable sets and the different levels of infinity.

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