Let's solve Exercise 4 step by step.

Given:

Let $X$ be a set and $A \subseteq X$. We define $A^c = \{x \in X : x \notin A \}$ as the complement of $A$ in $X$.

a) Give $\emptyset^c$ and $X^c$.

• $\emptyset^c$: The complement of the empty set in $X$ is $X$ because everything in $X$ is not in $\emptyset$. So, $\emptyset^c = X$.
• $X^c$: The complement of $X$ in $X$ is $\emptyset$, as there are no elements in $X$ that are not in $X$. Therefore, $X^c = \emptyset$.

b) Show that $(A^c)^c = A$ for every subset ( A \subseteq X \