Q.Verify the distributive law $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ using $A=\{1,2,3,4\}$, $B=\{3,4,5\}$, $C=\{4,5,6\}$.

Compute the left side by forming the union first and then intersecting with $A$; compute the right side by intersecting separately and then taking the union; confirm they match.

Step 1 — Compute the LHS. $B \cup C = \{3,4,5,6\}$. Then $A \cap (B\cup C) = \{1,2,3,4\} \cap \{3,4,5,6\} = \{3,4\}$.

Step 2 — Compute the RHS. $A \cap B = \{1,2,3,4\} \cap \{3,4,5\} = \{3,4\}$. $A \cap C = \{1,2,3,4\} \cap \{4,5,6\} = \{4\}$. Their union: $(A\cap B) \cup (A\cap C) = \{3,4\} \cup \{4\} = \{3,4\}$. …

Computing $A \cap B \cap C$ instead of the required union $(A\cap B)\cup(A\cap C)$ on the right-hand side — the distributive law's RHS …