Exercises
[05R]Let \(X\) be a non-empty set, and \(A ⊆ X\). We will denote with \(A^ c=X⧵ A=\{ x∈ X:x∉ A\} \) the complement of \(A\) in \(X\).
We define the characteristic function \({\mathbb 1}_ A: X→ℤ\) by
\[ {\mathbb 1}_ A(x)= \begin{cases} 1 & \text{if}~ x∈ A\\ 0 & \text{if}~ x∉ A\\ \end{cases} ~ . \]Prove that
\[ {\mathbb 1}_{A^ c} = 1-{\mathbb 1}_ A ~ ~ ,~ ~ {\mathbb 1}_{A∩ B} = {\mathbb 1}_ A {\mathbb 1}_ B ~ ~ ,~ ~ {\mathbb 1}_{A∪ B} = {\mathbb 1}_ A +{\mathbb 1}_ B - {\mathbb 1}_ A {\mathbb 1}_ B \]