Lesson 8

A set-theoretic identity is an equality that holds for all sets involved. For example, \[(A \cap B) \setminus C = (A \setminus C) \cap B \ .\] One can verify that this is indeed an identity in various ways:

1. Prove both inclusions. Let us prove that \((A \cap B) \setminus C \subseteq (A \setminus C) \cap B\). Assume \(x \in (A \cap B) \setminus C\). This means \(x \in A, x \in B, x \not \in C\). Then \(x \in A \setminus C\), and therefore \(x \in (A \setminus C) \cap B\). Now let us prove that \((A \setminus C) \cap B \subseteq (A \cap B) \setminus C\). Assume \(x \in (A \setminus C) \cap B\). This means \(x \in (A \setminus C)\) and \(x \in B\). Then \(x \in A\) and \(x \not \in C\). Consequently, \(x \in (A \cap B) \setminus C\).
2. One may write down “truth tables” of both expressions and verify that they match.

   

3. Draw a Venn diagram and verify that both expressions define the same region. (When there are more than three sets, this becomes less convenient.)