For a closed economy, it is impossible for the GDP deflator and the CPI to move into opposite directions (for example, the GDP deflator increases while the CPI falls). Is this true or false and why?

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I think the answer is this... but I'm not too sure. There are mainly two reasons the CPI and the GDP deflator move differently. First of all, the reason by the difference of the target affects this gap widely, besides the other reason such as the formula affects this gap, too.