1. **Problem Statement:** Prove that the intersection of sets $A$ and $B$ is commutative, i.e., $A \cap B = B \cap A$. 2. **Definition of Intersection:** The intersection of two sets $A$ and $B$, denoted $A \cap B$, is the set of all elements that are in both $A$ and $B$. Formally, $$x \in A \cap B \iff x \in A \text{ and } x \in B.$$ Similarly, $$x \in B \cap A \iff x \in B \text{ and } x \in A.$$ 3. **Proof:** To prove $A \cap B = B \cap A$, we need to show two inclusions: - If $x \in A \cap B$, then $x \in B \cap A$. - If $x \in B \cap A$, then $x \in A \cap B$. 4. **First inclusion:** Suppose $x \in A \cap B$. By definition, $x \in A$ and $x \in B$. Since "and" is commutative, $x \in B$ and $x \in A$, so $x \in B \cap A$. 5. **Second inclusion:** Suppose $x \in B \cap A$. By definition, $x \in B$ and $x \in A$. By commutativity of "and", $x \in A$ and $x \in B$, so $x \in A \cap B$. 6. **Conclusion:** Since both inclusions hold, we conclude $$A \cap B = B \cap A.$$ This shows the intersection operation is commutative.