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So i'm relatively new to computer science and have been learning set theory and am stumped on a question in it.

The question specifies that we're only looking at subsets of universe U = {0,...,n-1}. For a set S ⊆ U, the characteristic vector denoted by Xs is an n-bit string where j is 1 if and only if j ∈ S. Example, n = 4, S = {1, 3} Xs = 1010.

it is asking what is XØ and XU. I think I am completely overthinking this but I can't figure out what XØ and XU are referring to.

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Characteristic vector is a string representation of a set in this case. For a set $S$, the characteristic vector is denoted by $X_S$. $X_\emptyset$ and $X_U$ are referring to the characteristic vector of $U$ (Universal set) and $\emptyset$ (Empty set) respectively. In your example, $$\text{If }n = 4 \text{ and } S = \{1, 3\} \text{, then } X_s = 1010.$$ We simply need to replace $S$ with a set $\{0, 1, 2, \dots n-1\}$ to find $X_U$ and with $\{\} = \emptyset$ to find $X_\emptyset$. I think you can leave $n$ as a variable unless it is fixed to some value.

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