How to Prove the Cardinality of Unions of Infinite Sets?

In summary, to prove that the union of c sets of cardinality c has cardinality c, we can use a one-to-one and onto function to map the union of c intervals to the real numbers. This can be demonstrated by showing that a countable union of countable sets is countable. By associating the c-union of c-sets with R2, we can establish a bijection between R and R2, thus proving the statement. It may take some time to grasp the concept of infinity, but with persistence and guidance, it can be understood.
  • #1
mufq15
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Homework Statement


Prove that the union of c sets of cardinality c has cardinality c.


Homework Equations





The Attempt at a Solution


Well, I could look for a one-to-one and onto function... maybe mapping the union of c intervaks to the reals, or something? I know how to demonstrate that a countable union of countable sets is countable, by showing how to label them.
I'm having a hard time with this one, though.
 
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  • #2
[tex]\mathbb{R}^2 = \bigcup _{r \in \mathbb{R}} (\mathbb{R} \times \{ r\} )[/tex]

This should give you an easy way to associate a c-union of c-sets with R2. Now all you need is a bijection between R and R2.
 
  • #3
Ohh, I think I finally get it! (after thinking about it for a loong while...) Infinity is hard for me to wrap my head around. Thanks a lot for your help.
 

FAQ: How to Prove the Cardinality of Unions of Infinite Sets?

What is the concept of "cardinality" in mathematics?

Cardinality refers to the size or number of elements in a set. In mathematics, it is used to compare the sizes of different sets and determine if they have the same number of elements or if one set is larger than the other.

What are "infinite sets"?

Infinite sets are sets that have an unlimited number of elements. This means that no matter how many elements you count, there will always be more elements in the set.

Can infinite sets have different sizes?

Yes, infinite sets can have different sizes. This is determined by the concept of "cardinality", which compares the sizes of sets. Some infinite sets may have the same cardinality, meaning they have the same number of elements, while others may have different cardinalities, meaning one set has more elements than the other.

What is the "Cantor's diagonal argument" and how does it relate to the cardinality of infinite sets?

Cantor's diagonal argument is a mathematical proof that shows the cardinality of the set of real numbers is larger than the cardinality of the set of natural numbers. This means that there are more real numbers than natural numbers, even though both sets are infinite. The argument involves creating a new number that is not in the original set by manipulating the digits in the numbers in the set.

Do all infinite sets have the same cardinality?

No, not all infinite sets have the same cardinality. As mentioned before, the cardinality of a set is determined by the number of elements in the set. Since there is no limit to the number of elements in an infinite set, there can be infinite sets with different cardinalities. For example, the set of natural numbers has a different cardinality than the set of real numbers, even though both sets are infinite.

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