Proving a Set Is Empty: Disjoint Sets and Element Chasing

State the definition that $A$ and $B$ are disjoint provided $A \cap B = \emptyset$, and sketch the Venn diagram with $A \subseteq B$ showing $A$ and $B^c$ disjoint.

Write the contrapositive outline (assume $T \ne \emptyset$, prove $\neg P$) and the contradiction outline (assume $P$ and $T \ne \emptyset$), noting that both give an element of $T$ to work with.

Write out both directions: the forward one via its contrapositive starting from $x \in A \cap B^c$, the reverse one by choosing $x \in A$ and ruling out $x \notin B$.

Prove $\{x \in \mathbb{Z} \mid x \equiv 3 \pmod{12}\}$ and $\{y \in \mathbb{Z} \mid y \equiv 2 \pmod{8}\}$ are disjoint by contradiction through $1 = 8n - 12m$, and define the term element-chasing proof.