Choose-an-Element Proofs: Multiples of 6 Are a Subset of the Evens

State when it is used (a universal quantifier such as $(\forall x \in \mathbb{Z})[(x \in S) \to (x \in T)]$ in the backward process) and record its rule: name an arbitrary element and assume only its defining property.

Write $S$ and $T$ in set builder notation, then write the full proof from "Let $x \in S$" through $x = 6m = 2(3m)$ with $3m \in \mathbb{Z}$ to the conclusion $x \in T$.

Record that $S \subsetneq T$ means $S \subseteq T$ and $S \ne T$, show $S \ne T$ with the element $2 \in T$, $2 \notin S$, and redo both steps for multiples of 9 and multiples of 3.

Draw and shade the diagrams suggesting $A \subseteq B$ gives $B^c \subseteq A^c$ and $A - (A - B) = A \cap B$, and note that each relationship still needs a proof.