$[$ Mathematical logic (miscellaneous). $]$

Three friends - Petr, Roman, and Sergey - study at the mathematics, physics, and chemistry faculties. If Petr is a mathematician, then Sergey is not a physicist. If Roman is not a physicist, then Petr is a mathematician. If Sergey is not a mathematician, then Roman is a chemist. Can you determine the specialization of each?

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Think about whether Roman can be a mathematician. Recall problems 71, 132, 144, 189, 195.

## Solution

Assume that Roman is not a physicist, then (by condition 2) Peter is a mathematician, but if Peter is a mathematician, then Sergey (by condition 1) is not a physicist - this is a clear contradiction. Therefore, Roman is a physicist. Then Sergey is a mathematician - otherwise (by condition 3) Roman would be a chemist. Therefore, Peter is a chemist. Thus: Peter is a chemist, Roman is a physicist, Sergey is a mathematician.

## Answer

Peter is a chemist, Roman is a physicist, Sergey is a mathematician.

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