Q.If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: A∩(B∪C) = (A∩B) ∪ (A∩C).

$B \cup C = \{3,4,5,6\}\cup\{4,5,6,7,8\} = \{3,4,5,6,7,8\}$. Left side: $A \cap (B\cup C) = \{1,2,3,4\}\cap\{3,4,5,6,7,8\} = \{3,4\}$. Right side: $A \cap B = \{1,2,3,4\}\cap\{3,4,5,6\} = \{3,4\}$ and $A \cap C = \{1,2,3,4\}\cap\{4,5,6,7,8\} = \{4\}$, so $(A\cap B)\cup(A\cap C) = \{3,4\}\cup\{4\} = \{3,4\}$. Both sides equal {3,4}, verifying the distributive law of intersection over union (this is the correct distributive identity …

Compute B∪C first, intersect with A for the left side; separately compute A∩B and A∩C, then …

The printed statement in the source has a misprint (it shows an extra ∪ instead of ∩ on the right); always use the correct distributive law $A\cap(B\cup C)=(A\cap B)\cup(A\cap C) …