Transitivity property not clear.

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The discussion centers on the confusion regarding the transitivity property of equivalence relations as defined on Wikipedia. The transitivity property states that for any three elements a, b, and c, if a is related to b and b is related to c, then a must be related to c. The participant expresses difficulty in seeing this property illustrated in an example. However, it is clarified that examining all combinations of a, b, and c where the relations hold shows that a is indeed related to c in every case. This confirms the validity of the transitivity property in equivalence relations.
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i am reading equivalence relation on wikipedia. and in this examply. i don't see the transitity property according to the definition of transitivity which is: For every three elements a, b, and c in X, if a ~ b and b ~ c, then a ~ c (transitivity).

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any explanation please?
 
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If you go through all possible combinations of a, b, and c for which a~b and b~c hold, you will find that a~c also holds in all these cases.
 

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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