Prove that the empty set is the zero element for the Cartesian product:

$$A \times \emptyset = \emptyset \times A = \emptyset$$Prove that the Cartesian product is anticommutative:

$$A \times B = B \times A \longleftrightarrow A = B$$Prove that the Cartesian product distributes over intersection:

$$\begin{align} & (A \cap B) \times C = (A \times C) \cap (B \times C) \\ & C \times (A \cap B) = (C \times A) \cap (C \times B) \end{align}$$Prove that the Cartesian product distributes over union:

$$\begin{align} & (A \cup B) \times C = (A \times C) \cup (B \times C) \\ & C \times (A \cup B) = (C \times A) \cup (C \times B) \end{align}$$