1. Foundations of Proof

The language of analysis, from the ground up — propositions and quantifiers through direct proof, contrapositive, contradiction, strong induction, equivalence relations, and function proofs, following the arc of Hammack’s Book of Proof.

• Propositions and Truth Tables — Switch the connective and watch the new column fill in while the $P$, $Q$ columns stay fixed. Compare the shaded $P \to Q$ column with its contrapositive $\lnot Q \to \lnot P$ row by row — identical in all four rows. The converse $Q \to P$, though, splits off in the $(T, F)$ row.
• Universal and Existential Quantifiers — A game on the number line: the challenger names $x$, and you answer with the bigger point $y = x + 1$ — $\forall x \,\exists y \,(y > x)$. Swap the quantifier order and the same game collapses. The $\varepsilon$-$\delta$ line is exactly this pattern, with $\delta = \varepsilon/2$ answering $f(t) = 2t$.
• Direct Proof — The sum of two rationals $r_1 = \frac{p_1}{q_1}$, $r_2 = \frac{p_2}{q_2}$ on a number line, with the computation $\frac{p_1 q_2 + p_2 q_1}{q_1 q_2}$ running straight from hypothesis to conclusion — that is a direct proof.
• Proof by Contrapositive — The $n = 1..12$ grid where the $n^2$-even rows are exactly the $n$-even rows — where the direct proof of "$n^2$ even $\Rightarrow$ $n$ even" stalls, the contrapositive "odd $n$ $\Rightarrow$ odd $n^2$" opens up through $(2k+1)^2 = 2(2k^2+2k)+1$.
• Proof by Contradiction — Suppose the primes $p_1, \ldots, p_k$ were all there are — the chips show the list, $P = p_1 \cdots p_k$, and $N = P+1$. $N$ is divisible by no prime on the list (remainder 1) yet has a prime factor, and that factor is off the list — a contradiction. At $k = 6$, $N = 30031 = 59 \times 509$ is composite and the proof still stands, teaching the shape of contradiction (assume $\lnot P$ → contradiction → $P$).
• Proof by Cases and Existence Proofs — Moving the point $(x, y)$ shades its quadrant and unfolds that case's computation of $|xy| = |x|\cdot|y|$ — the four cases (and the axes) cover the whole plane. And from $c = \sqrt{2}^{\sqrt{2}}$ two branches fork and rejoin at "irrational $a, b$ exist with $a^b$ rational" without ever revealing which branch is true — the flavor of non-constructive existence.
• Counterexamples and If and Only If — Watch a single counterexample demolish a universal claim, tested on four false claims — every prime is odd, $4 \mid n^2 \Rightarrow 4 \mid n$, sums of irrationals are irrational, $|x| = |y| \Rightarrow x = y$. Slide $n$ and flip the hypothesis and conclusion lights until $\lnot(\forall x\, P(x)) = \exists x\, \lnot P(x)$ feels mechanical. The mirror skill — proving each arrow of $P \iff Q$ separately — is carried out in the problems.
• Strong Induction — Draw the factorization tree of $n = 30$ and strong induction explains itself — the splits drop to much smaller numbers like $15$ and $2$, not to $n - 1 = 29$. Slide $n$, watch the tree redraw, and follow the induction step that assumes all of $P(2), \dots, P(n-1)$ at once; the weak hypothesis $P(n-1)$ alone can never reach the factors.
• Equivalence Relations — Congruence modulo $n$ as an equivalence relation: the three axioms — reflexive, symmetric, transitive — are brute-force verified on every element, pair, and triple of $U = \{0, \dots, 3n-1\}$, and the classes $[r] = \{r, r+n, r+2n\}$ visibly partition the set as $n$ slides.
• Proofs about Composite Functions — Proofs about composite functions, tested on concrete tables: $g \circ f$ injective forces $f$ injective, $g \circ f$ surjective forces $g$ surjective — and the arrow diagrams show exactly why the converses fail ($g$ need not be injective, $f$ need not be surjective). For bijections, $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$ is computed out loud.