I am trying to understand what a disjoint decomposition is, specifically in the following proof of Example IV.8 (the last paragraph):
I don't understand how they got the last line of Example IV.8 from $(\dagger)$.
2026-05-16 09:43:51.1778924631
Trying to understand a proof where disjoint decomposition of $\mathbb R^n$ is used.
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$\mathbb{R}^n = X$ is the disjoint union of the given sets, say $A, B, C$. That is $X = A \cup B \cup C$. Nothing that is in C can be in $A$ or $B$ (sets are disjoint), so $C \subseteq (A \cup B)^c$. Also if something is not neither present in $A$ nor in $B$, it must be in $C$, as their union is $X$. So $(A \cup B)^c \subseteq C$. Together, we get $C = (A \cup B)^c$.