If A,B denumerable prove AUB denumerable

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In summary, to prove that AUB is denumerable given that A and B are denumerable sets, it is necessary to show that there exists a bijection between AUB and the set of natural numbers. This can be achieved by defining a function h(n) in terms of bijections f:N-->A and g:N-->B, and using the fact that n->2n is an injective function to show that the composition of injective functions is also injective.
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jsmith0476
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Homework Statement


Prove that if A and B are denumerable sets, not necessarily disjoint, then AUB is denumerable.


Homework Equations





The Attempt at a Solution


So I started my saying that since A and B are denumerable then, f:N-->A and g:N-->B are bijections. The I defined h(n) in terms of f and g as such:
h(n) = f(n/2) if n is even
h(n) = g((n+1)/2) if n is odd
This is all I have so far. I am not sure where to go or what this tells me.
 
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  • #2
Try mapping the sets to the naturals, the maps have to be injective. A hint is that n->2n is an injection and composition of injective functions is injective.
 

FAQ: If A,B denumerable prove AUB denumerable

What does "denumerable" mean in this context?

"Denumerable" refers to a set that can be counted or put in a one-to-one correspondence with the natural numbers (1, 2, 3, ...). This means that each element in the set can be associated with a unique natural number.

How do you prove that two sets are denumerable?

To prove that two sets, A and B, are denumerable, you must show that there exists a one-to-one correspondence between the elements of A and B. This means that each element in A must be paired with a unique element in B, and vice versa.

Can you give an example of a denumerable set?

An example of a denumerable set is the set of positive even numbers. Each even number can be paired with a unique natural number (e.g. 2 with 1, 4 with 2, 6 with 3, etc.), making it denumerable.

How does proving that AUB is denumerable imply that both A and B are denumerable?

If AUB is denumerable, it means that there exists a one-to-one correspondence between the elements of AUB and the natural numbers. This implies that both A and B are denumerable because A and B are subsets of AUB, and any subset of a denumerable set is also denumerable.

Are there any other ways to prove that AUB is denumerable?

Yes, there are other ways to prove that AUB is denumerable. One way is to show that AUB is countable, meaning that the elements of AUB can be listed in a sequence. Another way is to show that AUB is infinite, meaning that it has an infinite number of elements. Both of these properties imply that AUB is denumerable.

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