MHB Does cardinality of a set refer to the number of elements it has?

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Cardinality of a set refers to the number of elements within that set, defining its size. While this concept applies straightforwardly to finite sets, it becomes more complex when discussing infinite sets. The discussion emphasizes that cardinality is not just a reference to quantity but represents that specific number. Understanding cardinality is essential for distinguishing between different types of infinite sets. Overall, cardinality serves as a fundamental concept in set theory.
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Is cardnality of a set refers to the number of elements that set has?
 
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yakin said:
Is cardnality of a set refers to the number of elements that set has?
It does not only refer to the number of elements; it is that number. :) For infinite sets, though, it is more complicated.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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