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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 \

Soit X un ensemble et A ⊆ X. Notons Ac = {x ∈ X : x ∉ A} le complémentaire de A dans X. a) Donner ∅c et Xc. b) Montrer que (Ac)c = A pour toute partie A ⊆ X. c) Montrer que X = A ∪ Ac et que A ∩ Ac = ∅ pour tout A ⊆ X. d) Montrer que si A ⊆ B ⊆ X alors Bc ⊆ Ac. e) Montrer que (A ∩ B)c = Ac ∪ Bc et que (A ∪ B)c = Ac ∩ Bc pour toutes parties A, B ⊆ X.

Solve problems involving set theory with complements, including proving properties like (Ac)c = A, union and intersection of sets, and De Morgan's laws. This problem is ideal for students learning about set theory and complements at an undergraduate level.

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