Simple Proof Methods

On slide 56 of Topic 2, we presented a table with $19$ set identities (= equivalences). It's time to show how to prove some! But first, here's a definition which we already mentioned briefly in Topic 2:

Definition. [Set equality] Two sets $S_1$ and $S_2$ are called equal if $S_1 \subseteq S_2$ and $S_2 \subseteq S_1$. We write: $S_1 = S_2$.

Each of the set identities is provable, so we can label them as theorems. Let's do it for the following identities:

1. $A \cap U = A$ (One of the Identity Laws)
2. $A \cap \emptyset = \emptyset$ (One of the Null Laws)
3. $A \cup B = B \cup A$ (One of the Commutative Laws)
4. $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ (One of the Distributive Law)
5. $A \cup (A \cap B) = A$ (One of the Absorption Law)