\(A \times (B\cup C) = (A\times B)\cup (A\times C)\)
Let \((x, y) \in A\times (B \cup C)\)
\(\Leftrightarrow x \in A \) and \(y \in (B\cup C)\)
\(\Leftrightarrow x \in A \) and \(y \in B \;or\;y\in C\)
\(\Leftrightarrow x \in A \) and \(y \in B \;or\;x\in A \;or\; y\in C\)
\(\Leftrightarrow (x,y) \in A \times B \;or\; (x,y)\in (A \times C)\)
\(\Leftrightarrow (x, y) \in (A \times B)\cup (A\times C)\)
\(\therefore A \times (B \cup C) = (A \times B)\cup (A \times C)\)
