Definition 2.4.7.1.1 (01JU): The Internal Hom of a Powerset

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The internal Hom of $P (X)$ from $U$ to $V$ is the subset $[U, V]_{X}$ ¹ of $X$ defined by

\begin{aligned} [U, V]_{X} & \overset{def}{=} U^{c} \cup V \\ = (U ∖ V)^{c} \end{aligned}

where $U^{c}$ is the complement of $U$ of Definition 2.3.11.1.1.

¹Further Notation: Also written

{Hom}_{P (X)} (U, V)

Proof of Definition 2.4.7.1.1.

We have

\begin{aligned} (U ∖ V)^{c} & \overset{def}{=} X ∖ (U ∖ V) \\ = (X \cap V) \cup (X ∖ U) \\ = V \cup (X ∖ U) \\ \overset{def}{=} V \cup U^{c} \\ = U^{c} \cup V, \end{aligned}

where we have used:

This finishes the proof.

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