The indicator function of a set \(A \subset X\) is defined by

\[\mathbb{1}_{A}(x):= \begin{cases}1 & \text { if } x \in A \\ 0 & \text { if } x otin A\end{cases}\]

Check that for \(A, B, A_{i} \subset X, i \in I\) (arbitrary index set) the following equalities hold:

(i) \(\quad \mathbb{1}_{A \cap B}=\mathbb{1}_{A} \mathbb{1}_{B}\)

(ii) \(\quad \mathbb{1}_{A \cup B}=\min \left\{\mathbb{1}_{A}+\mathbb{1}_{B}, 1ight\}\);

(iii) \(\mathbb{1}_{A \backslash B}=\mathbb{1}_{A}-\mathbb{1}_{A \cap B}\);

(iv) \(\mathbb{1}_{A \cup B}=\mathbb{1}_{A}+\mathbb{1}_{B}-\mathbb{1}_{A \cap B}\);

(v) \(\mathbb{1}_{A \cup B}=\max \left\{\mathbb{1}_{A}, \mathbb{1}_{B}ight\}\);

(vi) \(\quad \mathbb{1}_{A \cap B}=\min \left\{\mathbb{1}_{A}, \mathbb{1}_{B}ight\}\);

(vii) \(\quad \mathbb{1}_{\bigcup_{i \in I} A_{i}}=\sup _{i \in I} \mathbb{1}_{A_{i}}\);

(viii) \(\mathbb{1}_{\bigcap_{i \in I} A_{i}}=\inf _{i \in I} \mathbb{1}_{A_{i}}\).