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 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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