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Edward357
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How do I prove that the set of all finite subsets of an infinite set has the same cardinality as that infinite set?
Edward357 said:How do I prove that the set of all finite subsets of an infinite set has the same cardinality as that infinite set?
Careful. The OP wants to prove that the set of all finite subsets of X has the same cardinality as X. This is certainly true whenever X is infinite.Focus said:You can't because its not true. Take the reals which have aleph-one cardinality, the subset {1} has cardinality 1 which is not aleph-one. Same goes for infinite subsets, naturals are a subset of reals but naturals have cardinality aleph-null.
Focus said:You can't because its not true. Take the reals which have aleph-one cardinality, the subset {1} has cardinality 1 which is not aleph-one. Same goes for infinite subsets, naturals are a subset of reals but naturals have cardinality aleph-null.
morphism said:What have you tried?
Your proof looks right, including this assertion here as well. But from the way you phrased it, I'm not entirely sure your justification for this assertion is correct. Would you elaborate?The union of all K^n is equivalent to lKl + lKl with the operation repeated countably many times. The sum must have cardinality lKl.
Focus said:Sorry I misread the post. You could try it this way (assuming CH): you know that if S is the set of all finite sets then [tex] a_i \in X \qquad \{\{a_1\},\{a_2\},...\} \subset S[/tex] thus the cardinality of S is greater of equal to the cardinality of X. Now S is a subset of P(X) thus the cardinality of S is less than or equal to [tex] 2^{|X|} [/tex]. I think you may be able to construct a proof to make that last inequality strict (perhaps by Cantors diagonal argument), then you have that |S|=|X|.
Hope this is more helpful
The cardinality of this set is equal to the cardinality of the infinite set itself. This means that the number of elements in the set of all finite subsets is the same as the number of elements in the infinite set.
This is due to the concept of "counting" infinite sets. While we may think that an infinite set has an infinite number of elements, certain infinite sets can be counted in a similar way to finite sets. In this case, the infinite set and the set of finite subsets have the same number of elements, making their cardinalities equal.
No, the set of all finite subsets is not infinite. It has a finite number of elements, even though it may seem like it would have an infinite number due to being a subset of an infinite set.
No, the cardinality of the set of all finite subsets can never be greater than the cardinality of the infinite set. This is because the set of finite subsets is a subset of the infinite set and therefore cannot have more elements than the infinite set.
The concept of "counting" infinite sets and the equal cardinality of the set of finite subsets and infinite sets have significant implications in the study of infinite sets. It allows for the comparison and classification of different types of infinite sets, as well as the understanding of the size and structure of infinite sets.