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.
 
There is a nice little variation of the problem. The host says, after you have chosen the door, that you can change your guess, but to sweeten the deal, he says you can choose the two other doors, if you wish. This proposition is a no brainer, however before you are quick enough to accept it, the host opens one of the two doors and it is empty. In this version you really want to change your pick, but at the same time ask yourself is the host impartial and does that change anything. The host...

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