EE 342

In all the following, \(\Omega\) is a set, and all other sets are subsets of \(\Omega\). All indicator functions are functions from \(\Omega\) to \(\{0,1\}\).

  1. Prove that union/intersections distribute over each other, namely \(S \cap (T \cup U) = (S \cap T) \cup (S \cap U)\) and \(S \cup (T \cap U) = (S \cup T) \cap (S \cup U)\).

  2. Prove that \((\bigcup_n S_n)^c = \bigcap_n S_n^c\), and that \((\bigcap_n S_n)^c = \bigcup_n S_n^c\).

  3. Can you show that
    • If \(A \subset B\), then \({\mathbf 1}_A \le {\mathbf 1}_B\)
    • If \(A\) and \(B\) are disjoint, then \({\mathbf 1}_{A\cup B}={\mathbf 1}_A + {\mathbf 1}_B\)
    • For any \(A\) and \(B\), \({\mathbf 1}_{A\cup B} \le {\mathbf 1}_A + {\mathbf 1}_B\)
  4. Let \(A_1,\ldots, A_n\) be subsets of \(\Omega\). Can you express the indicator functions \({\mathbf 1}_{A_1\cap A_2\cap \cdots A_n}\) and \({\mathbf 1}_{A_1\cup A_2\cup \cdots A_n}\) in terms of the indicator functions \({\mathbf 1}_{A_1}\), \({\mathbf 1}_{A_2}\) \(\cdots\) \({\mathbf 1}_{A_n}\)?