True or false statements with sets

Question

How can I do the following exercise?

$A$, $B$ and $C$ are arbitrary sets. Motivating appropriately the answers with proofs or counter-examples, establish whether the following statements are true or false:

• $A \cap (B \cup C) \subseteq (A \cap B) \cup C$;
• $A \cap (B \cup C) \supseteq (A \cap B) \cup C$;
• $A \cap (B \cup C) = (A \cap B) \cup C$.

Answer 1

First of all, we know that $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

• For the first one, it claims $(A \cap B) \cup (A \cap C) \subseteq (A \cap B) \cup C$. This is true because we always have $A \cap C \subseteq C$.

• For the second one, it claims $(A \cap B) \cup (A \cap C) ⊇ (A \cap B) \cup C$, which is false when $A \cap C \ne C$ (Take $A = \{1,2\}$, $B = \{1,3\}$, $C = \{2,3\}$ as a counter-example)

• For the third one, we can use the same counter-example given in the second one to show that the statement is false.