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In set $T = \{(0+2)^{i+5}|\space i \in \mathbb{N}\}$, what is the meaning of the $(0+2)^{i+5}$?

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    i plan on marking the answer when i get my pset back :)2012-09-05

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A likely possibility, given your comment, is that $(0+2)$ is to be interpreted as a regular expression, namely "either 0 or 2" and the apparent exponentiation would be read as concatenation of the elements ({0, 2}) of that set. Under this reading, the set $T$ would be (assuming $\mathbb{N}$ doesn't include zero) $\{000000, 022222,0202020200002, , \dots\}$, namely all strings of $0$s and $2$s of length greater than or equal to 6.

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(For me $0 \in \mathbb{N}$: adjust to personal taste)

I would read $\{(0+2)^{i+5}|\space i \in \mathbb{N}\}$ as $\{(0+2)^{0+5}, (0+2)^{1+5}, (0+2)^{2+5}, (0+2)^{3+5}, \ldots\}$, i.e. as $\{32,64,128,256 \ldots\}$.

I might wonder why the writer did not just say $\{2^{i+5}|\space i \in \mathbb{N}\}$ but I would not worry too much.

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    Probably because, as James commented, the problem is looking at digits, rather than their values as expressions as numbers.2012-08-31