MHB Equivalences and Partitions and Properties of binary relations

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The discussion centers on a request for assistance with questions about equivalences, partitions, and properties of binary relations. The response emphasizes the importance of adhering to forum rules, particularly regarding the number of questions allowed per thread. Rule 8 specifically limits users to two questions, while the original post included twenty. Participants are encouraged to familiarize themselves with the forum's guidelines to ensure productive interactions. Following the rules will enhance the likelihood of receiving help effectively.
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If someone could explain some of the steps needed to work out these 2 questions it would be much appreciated!

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Hello, sadsadsadsa!

We would be glad to help, but you should study the http://mathhelpboards.com/rules/ first. On the rules page, click on the Expand button in the top-left corner. In particular, read rules 8, 11 and 6. Thus, rule 8 asks not to ask more than two questions in a thread, while you posted 20 questions in total.
 

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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