Cardinality of the Union of Two Sets that have Same Cardinality as Real Numbers

In summary, to show that U \cup V has the same cardinality as R, it is sufficient to show that there exists a bijective map from R to U \cup V. This can be done by first showing the existence of bijective maps from R to U and from R to V, and then composing them to get a bijective map from R to U \cup V.
  • #1
georgetown13
7
0

Homework Statement


Let U and V both have the same cardinality as R (the real numbers). Show that U[tex]\cup[/tex]V also has the same cardinality as R.

Homework Equations


The Attempt at a Solution


Because U and V both have the same cardinality as R, I that that this means
[tex]\exists[/tex] f: R[tex]\rightarrow[/tex]U that is one-to-one and onto.
[tex]\exists[/tex] g: R [tex]\rightarrow[/tex] V that is one-to-one and onto.

I think I need to show that [tex]\exists[/tex] h: R [tex]\rightarrow[/tex] U [tex]\cup[/tex] V.

But how do I get to that point? Please help! I would greatly appreciate any assistance.
 
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  • #2
Do you know that, for example, that (-infinity,0] and (0,infinity) both have the same cardinality as R?
 
  • #3
Dick said:
Do you know that, for example, that (-infinity,0] and (0,infinity) both have the same cardinality as R?

Yes, but I can't just provide an example to prove the statement, right?

I understand the general concepts behind this proof but am having a difficult time putting it down in mathematical terms.
 
  • #4
georgetown13 said:
Yes, but I can't just provide an example to prove the statement, right?

I understand the general concepts behind this proof but am having a difficult time putting it down in mathematical terms.

I didn't mean it to be an example. In the nonmessy case where U and V are disjoint, then R maps bijectively to (-infinity,0]U(0,infinity) via the obvious map 'identity' then map (-infinity,0] bijectively to U and (0,infinity) bijectively to V. Compose them and you have a bijective map from R to U union V.
 

1. What is the definition of cardinality?

The cardinality of a set is the number of elements in that set. It represents the size or magnitude of a set.

2. How is the cardinality of a set determined?

The cardinality of a set can be determined by counting the number of elements in the set or by using a one-to-one correspondence with another set.

3. What does it mean for two sets to have the same cardinality?

If two sets have the same cardinality, it means that they have the same number of elements, regardless of the order or structure of the elements.

4. How is the cardinality of the union of two sets with the same cardinality as real numbers calculated?

The cardinality of the union of two sets with the same cardinality as real numbers is equal to the cardinality of the larger set. This is because the union of two sets includes all elements from both sets, and if both sets have the same cardinality, the larger set will still have the same number of elements after the union operation.

5. Can the cardinality of the union of two sets with the same cardinality as real numbers ever be less than the cardinality of real numbers?

No, the cardinality of the union of two sets with the same cardinality as real numbers will always be equal to the cardinality of real numbers because the union operation adds all elements from both sets without repetition, resulting in a larger set with the same number of elements.

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