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For example, given the characteristic function XA + XB - XA $ \cap $ B where A and B and subsets of set S.

How to find the subset of the characteristic function?

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Let $f = \chi_A + \chi_B - \chi_{A \cap B}$. Note that $f(x) = 1$ if and only if $$\chi_A(x) + \chi_B(x) = \chi_{A \cap B}(x) + 1$$ and, since each term is either $0$ or $1$, the latter happens if and only if the following cases holds:

  • $\chi_A(x) = 1$, $\chi_B(x) = 0$ and $\chi_{A \cap B}(x) = 0$.

  • $\chi_A(x) = 0$, $\chi_B(x) = 1$ and $\chi_{A \cap B}(x) = 0$.

  • $\chi_A(x) = \chi_B(x) = \chi_{A \cap B}(x) = 1$.

Now, the first two cases just tell us that $x$ must be in $A$ or $B$ but not in both; and of course, the latter tell us that $x$ must be in both. The only way that this happens is that $x$ is in $A \cup B$ (think about it, there is no other situation).

So, we prove that $f(x)=1$ if and only if $x \in A \cup B$. Hence, $f = \chi_{A \cup B}$.

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$X_A=X_{A\cap B}+X_{A\setminus B}$ and $X_B=X_{A\cap B}+X_{B\setminus A}$. Hence $X_A+X_B-X_{A\cap B}=X_{A\setminus B}+X_{B\setminus A}+X_{A\cap B}=X_{A\cup B}$

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