Q.Let $A = \{1, 2, 3\}$, $B = \{2, 3, 4\}$ and $C = \{3, 4, 5\}$. Verify the distributive property $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.

Compute the left-hand side and the right-hand side completely independently of each other, step by step, then compare the final sets.

Left-hand side: $A \cup (B \cap C)$.

• First, $B \cap C$: elements common to $B=\{2,3,4\}$ and $C=\{3,4,5\}$ are $3$ and $4$, so $B \cap C = \{3,4\}$.
• Then, $A \cup (B\cap C) = \{1,2,3\} \cup \{3,4\} = \{1,2,3,4\}$.

Right-hand side: $(A \cup B) \cap (A \cup C)$.

• First, $A \cup B = \{1,2,3\} \cup \{2,3,4\} = \{1,2,3,4\}$.
• Then, $A \cup C = \{1,2,3\} \cup \{3,4,5\} = \{1,2,3,4,5\}$.
• Finally, $(A\cup B) \cap (A\cup C) = \{1,2,3,4\} \cap \{1,2,3,4,5\} = \{1,2,3,4\}$ (every element of the smaller set $\{1,2,3,4\}$ is also in the larger one). …

Doing the operations in the wrong order — e.g. computing $A \cup B \cap C$ left-to-right without respecting that $(B \cap C)$ must be evaluated FIRST because of the brackets. Also, on the right-hand side, forgetting that BOTH $A\cup B$ and $A\cup C$ must be fully computed before intersecting them, rather than intersect …