# Episode 5: The Proof Laboratory

**Objective:** To solidify proof skills on the concrete yet abstract ground of sets and functions, and to master the combinatorial logic of counting and cardinality.

### I. Introduction: The Language of Structure

* **Sets as the Foundation:** Recognizing that every mathematical statement—from parity to primes—can be viewed as a claim about **set membership**.

* **The Proof Laboratory:** Transitioning from number-theoretic proofs to more abstract logical environments where **definitions** remain the primary "bricks".

### II. Segment 1: Set Theory Essentials (Chapter 13)

* **The Double-Inclusion Method:** Understanding that set equality A = B is a conjunction of two universal conditionals: A \subseteq B and B \subseteq A.

* **The Logic of Operations:** ...