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I have two questions about set notation

The first one is how you would write down a set of objects expressed by symbols? Let's say we have $n$ persons identified by $p_i$. Can I write: $$P = \{p_1, ..., p_n\}$$ for the complete set of persons?

I also saw the following notation: $$P = \{P_1, ..., P_n\}$$

Should I use uppercase letters for sets and lowercase indexed letters for objects of the set?

The second question is should I write $p_i \in P$ or $p \in P$?

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The set of persons can be denoted by $\{p_1,\ldots,p_n\}$. It doesn't matter if you use uppercase or lowercase letters, though often sets have uppercase and their elements have lowercase. You should write $p_i\in P$, not $p\in P$ as there is no $p$ to begin with.

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