Day 34 — Gate III

Updated

July 30, 2026

Day 34 — Gate III

Stage III · gate day
Goal: Closed-book mixed review of propositional logic, quantifiers, and proof methods (Days 23–33). Diagnose weak spots; retest before Stage IV.

Why this matters

Gates are mastery checks, not calendar markers. Stage IV (sets, relations, functions) assumes you can negate statements, choose a proof style, and not swap \(\forall\exists\). If this gate frays, repair now—later counting and graphs pile on justification load.

What this gate covers

Block	Days	Skills
Truth tables & connectives	23	Tables; tautology/contradiction; \(\rightarrow\) vacuous truth; XOR; precedence
Boolean algebra	24	De Morgan, absorption, simplify; DNF/CNF lite
Equivalence	25	Tables + law chains; contrapositive \(\equiv\); completeness awareness
Predicates & quantifiers	26	Translate; negate; restricted \(\forall\)/\(\exists\)
Nested quantifiers	27	Order matters; \(\exists!\); witnesses
Direct proof	28	Parity, divides
Contrapositive / contradiction	29	Method choice; \(\sqrt{2}\) shape
Cases & existence	30	Exhaustion; constructive existence
Weak / strong induction	31–32	\(P(n)\); loop invariant idea; structural lite
Fallacies	33	Affirming consequent; bad induction; quantifier shift

How to use this page

Closed book first pass (timebox \(90\)–\(120\) minutes).
Mark each item: ✅ confident / ⚠ shaky / ❌ failed.
Open notes only to repair ❌/⚠—then re-solve cold next day.
Do not proceed to Day 35 until you can clear a second pass on failed items.

Important

No programming labs. Optional calculator only to check arithmetic. Proofs are by hand.

Theory — rapid self-check sheet

Reproduce from memory on scrap paper, then compare:

Truth tables for \(p\rightarrow q\), \(p\leftrightarrow q\), \(p\oplus q\).
De Morgan laws (propositional) and quantifier negation rules.
\(p\rightarrow q \equiv \neg p \vee q \equiv \neg q \rightarrow \neg p\).
Restricted quantifiers: \(\forall x\in S\,P\) vs \(\exists x\in S\,P\).
Why \(\forall x\exists y\,P \not\equiv \exists y\forall x\,P\) (one counter-model).
Direct / contrapositive / contradiction templates.
Weak induction skeleton; strong when needed.
Loop invariant: init / maintain / exit+use.
Affirming the consequent in one line; illicit \(\forall\exists\) shift.
Definition of even/odd and \(a\mid b\).

Worked examples (gate warm-ups)

Warm-up A — Table
Show \((p \wedge q) \rightarrow p\) is a tautology (4 rows).

Warm-up B — Simplify
\(\neg(p \rightarrow \neg q) \equiv p \wedge q\) (named laws).

Warm-up C — Quantifiers
Negate \(\forall x \exists y\, (x < y)\) on \(\mathbb{Z}\) and state the truth of the original.

Warm-up D — Direct proof
If \(n\) odd then \(n^2\) odd.

Warm-up E — Contrapositive
If \(n^2\) even then \(n\) even.

Warm-up F — Induction
\(\sum_{i=1}^n i = n(n+1)/2\).

Warm-up G — Fallacy
Name: From “if admin then 2FA” and “has 2FA” infer “admin.”

Warm-up H — Cases
\(n(n+1)\) always even.

Warm-up I — Strong / postage lite
Why three consecutive bases when stepping by \(3\) in postage proofs?

Warm-up J — Nested English
Translate both orders: every student takes some course / some course taken by every student.

Gate exam (closed book)

Part I — Propositional (mix)

Full truth table for \((p \rightarrow q) \wedge (q \rightarrow p)\). Relate to \(\leftrightarrow\).
Is \((p \rightarrow q) \vee (q \rightarrow p)\) a tautology? Justify.
Simplify \(\neg(\neg p \wedge q)\) with named laws.
Prove or disprove equivalence: \(p \rightarrow (q \rightarrow r)\) vs \((p \rightarrow q) \rightarrow r\).
Write XOR in DNF and CNF.
Parenthesize under standard precedence: \(\neg p \vee q \rightarrow r\).
Sufficient vs necessary: one sentence each with arrows for “valid token / access.”
Show \(p \wedge (p \vee q) \equiv p\) (absorption) by algebra or table.

Part II — Quantifiers

Translate: “No banned user is admin.” Two equivalent forms.
Negate fully: \(\forall u\, (Auth(u) \rightarrow \exists r\, Role(u,r))\).
On \(\mathbb{R}\): truth of \(\forall x\exists y\,(xy=1)\) and \(\exists y\forall x\,(xy=1)\).
Expand \(\forall n\in\mathbb{N}\, P(n)\) and \(\exists n\in\mathbb{N}\, P(n)\) using \(\rightarrow\) / \(\wedge\).
Write \(\exists! x\, P(x)\) without the \(!\) symbol.
Free vs bound: mark variables in \(\forall x\, (P(x,y) \rightarrow \exists y\, Q(y))\).
Give a \(2\)-element model where \((\exists x\,P)\wedge(\exists x\,Q)\) but not \(\exists x\,(P\wedge Q)\).

Part III — Proofs (write full proofs)

Direct: If \(a\mid b\) and \(a\mid c\) then \(a\mid (b+c)\).
Contrapositive or contradiction: If \(n^2\) is odd then \(n\) is odd.
Contradiction classic: \(\sqrt{2}\) is irrational (full write-up).
Cases: For all \(n\in\mathbb{Z}\), \(n^2 \not\equiv 3 \pmod{4}\).
Weak induction: \(\sum_{i=0}^n 2^i = 2^{n+1}-1\) for \(n\ge 0\).
Weak or strong: \(3 \mid (4^n-1)\) for \(n\ge 0\).
Strong induction: Every integer \(n\ge 2\) has a prime factor.
Existence constructive: Exhibit witnesses for \(\exists x,y\in\mathbb{Z}\, (3x+5y=1)\) or prove impossible.
Biconditional: \(n\) even iff \(n+1\) odd (both directions).
Loop invariant sketch: For a loop that sets s += i for i=1..n, state \(I\), init, maintain, exit use.

Part IV — Fallacies (identify & refute)

Affirming the consequent—give formal shape and a CS-flavored counterexample.
Illicit \(\forall\exists \Rightarrow \exists\forall\)—counter-model on \(\mathbb{Z}\).
Proof by example for a universal claim—why invalid?
Horses fallacy—where does the step fail?
From \(p\rightarrow q\) infer \(\neg p \rightarrow \neg q\)—name and counter-assignment.
Circular reasoning vs legitimate inductive hypothesis—one paragraph.
False dichotomy in a case proof—invent a broken outline and fix it.

Part V — Mixed synthesis

Prove by induction on \(n\) that a set with \(n\) elements has \(2^n\) subsets (outline OK if full algebra clear).
Negate the \(\varepsilon\)-\(N\) limit definition form \(\forall\varepsilon>0\,\exists N\,\forall n\ge N\,(|a_n-L|<\varepsilon)\).
Choose methods: for each claim, pick direct / contra / cases / induction and justify in one line:
1. sum formula; (b) no largest integer; (c) \(n(n+1)\) even; (d) if \(6\mid n\) then \(2\mid n\).
Boolean: put \((p \rightarrow q) \wedge p\) into simplest equivalent form.
Translate sorted nondecreasing array of length \(n\) with nested quantifiers.
Explain vacuous truth of \(\forall x\in\emptyset\, P(x)\) in one sentence.
Functional completeness awareness: express \(p\vee q\) using only \(\neg\) and \(\wedge\).
Portfolio reflection: list three skills from Days 23–33 you will drill this week.

Scoring guide (self)

Score band	Meaning	Action
≥ 34/40 solid	Stage III ready	Proceed to Day 35
26–33	Patch holes	Retake failed numbers cold in 48h
≤ 25	Rebuild	Revisit Days 23–27 then 28–33; retake full gate

Count a multi-part proof item as split if needed (e.g. ½ for structure without algebra). Honesty > speed.

Optional deep extras (if clean early)

Prove Cassini’s Fibonacci identity for \(n\ge 1\) (state indexing).
Full loop invariant proof sketch for Euclid’s algorithm.
Show \(\{\mathrm{NAND}\}\) can express \(\neg\) and \(\wedge\) (formulas).
Prove \(\log_2 3\) irrational.
Structural induction: \(\mathrm{len}(xs{+\!\!+}ys)=\mathrm{len}(xs)+\mathrm{len}(ys)\).

CS connection (gate lens)

Specs are quantified implications; tests seek counterexamples to \(\forall\).
Refactors need \(\equiv\), not “looks similar.”
Invariants are induction; fallacies are production incidents waiting to happen.
Stage IV sets/relations will encode more of the same logic with \(\in\) and \(\subseteq\).

Common pitfalls (gate day)

Pitfall	What to do instead
Open book too early	Close it for pass 1
Sketchy proofs without witnesses	Exhibit \(k\) in parity/divisibility
Forgetting to negate nested quantifiers fully	Flip each quantifier
Method name without proof body	Write the proof
Skipping fallacy counterexamples	Always refute

Checkpoint

Self-check sheet reproduced from memory
Warm-ups A–J attempted cold
Gate exam Parts I–V scored
Failed items scheduled for retest
Three personal weak spots written down
Ready/not-ready decision for Stage IV

Two takeaways from Stage III:

Extended drill set (still closed book if possible)

Drill A — Tables under time pressure

Table for \(p\oplus q \oplus r\) (careful association).
Show \((p\rightarrow q)\rightarrow((\neg p\rightarrow q)\rightarrow q)\) is a tautology.
Contingency hunt: invent a 3-var formula with exactly 3 true rows.

Drill B — Algebra under time pressure

Simplify \(\neg(p\wedge\neg(q\vee p))\) fully.
Prove \(\neg(p\leftrightarrow q)\equiv p\leftrightarrow\neg q\).
Put \(p\rightarrow(q\wedge r)\) into CNF.

Drill C — Quantifier battery

Negate \(\exists x\forall y\exists z\, R(x,y,z)\).
Translate and negate: “Every nonempty file has a checksum.”
On \(\mathbb{N}\): \(\forall m\exists n\,(n>m)\) vs \(\exists n\forall m\,(n>m)\).
Write \(\exists!\) for “unique root of \(x^2-1=0\) in \(\mathbb{R}\)”—is it true? (No—two roots.)

Drill D — Proof battery

Direct: product of two odds is odd.
Contrapositive: if \(n^2\) not divisible by \(2\) then \(n\) not divisible by \(2\).
Contradiction: no largest negative integer? (false claim)—instead: no largest positive integer.
Cases: \(n^2\equiv 0\) or \(1\pmod 4\).
Induction: \(\sum_{i=1}^n (2i-1)=n^2\).
Strong: every \(n\ge 2\) composite or prime—trivial; instead every \(n\ge 2\) has prime factor.
Invariant: state \(I\) for while n > 0: n = n-1; c = c+1 counting down.

Drill E — Fallacy battery

58–62. Invent five one-line fallacious inferences and name them.

Repair plan template

Failed #	Topic day	Minutes drill	Retest date

Full sample solution outlines (check after attempt)

Item 18 outline (\(\sqrt{2}\)): assume \(p/q\) lowest terms → \(p^2=2q^2\) → \(p\) even → \(q\) even → contradict gcd.

Item 20 outline (geometric): base \(n=0\); IH sum to \(k\); add \(2^{k+1}\).

Item 22 outline (prime factor): strong IH on smaller factors if composite.

Item 26 outline (fallacy): \(P\rightarrow Q\), \(Q\), infer \(P\)—counter-model \(P\) false \(Q\) true.

Stage III exit criteria

You are ready for Stage IV when you can, cold:

Finish a 4-row and sketch an 8-row table.
Negate a two-quantifier sentence without hesitation.
Write direct, contrapositive, and induction proofs to a peer standard.
Name and refute three fallacies.

Timed mock (45 minutes)

Do only these, closed book, then stop:

#	Task	Minutes
M1	Table \((p\rightarrow q)\leftrightarrow(\neg q\rightarrow\neg p)\)	6
M2	Simplify \(\neg(p\vee\neg q)\wedge p\)	4
M3	Negate \(\forall x\exists y\,(P(x)\rightarrow Q(x,y))\)	5
M4	Direct: even+even=even	5
M5	Contrapositive: odd square ⇒ odd \(n\)	6
M6	Induction: \(\sum_{i=1}^n i = n(n+1)/2\)	8
M7	Name fallacy: \(P\rightarrow Q\), \(Q\), therefore \(P\)	2
M8	\(\forall\exists\) vs \(\exists\forall\) counter-model on \(\mathbb{Z}\)	5
M9	Cases: \(n^2\not\equiv 3\pmod 4\)	4

Pass bar: \(\ge 7/9\) solid. Below that: schedule full gate retake.

After mock reflection prompts

Which item was slowest?
Did you skip naming laws in algebra?
Did induction include an explicit \(P(n)\)?
Did quantifier negation flip every quantifier?

Second-pass protocol

Only reopen notes for ❌ items.
Write a correct solution once with notes.
Next day, redo ❌ cold without notes.
Log time-to-completion.

Stage III → IV teaser

You will reuse:

Element chase ≈ direct proof with \(x\in A\).
\(\subseteq\) proofs ≈ implication proofs.
Equivalence relations ≈ kernels of functions.
Induction on set size for \(|\mathcal{P}(S)|=2^n\).

Synthesis — Stage III portfolio

Gate III is not a new topic day; it is a compression of Days 23–33. Treat this synthesis as your one-page mental model before and after the exam.

Skill clusters (what “ready” means)

Cluster	Cold ability
Propositional	4–8 row tables; simplify with named laws; \(\rightarrow\) as \(\neg p\vee q\)
Quantifiers	Translate all/some/no; negate nested; \(\forall\exists\neq\exists\forall\)
Proof methods	Direct, contra, contrapositive, cases, weak/strong induction
Invariants	Init / maintain / exit+use in one paragraph
Fallacies	Name, counter-model, repair

Method choice drill (answer in one line each)

\(\sum_{i=1}^n i = n(n+1)/2\) →
If \(n^2\) even then \(n\) even →
\(n(n+1)\) always even →
\(\sqrt{2}\notin\mathbb{Q}\) →
No largest integer →
Every \(n\ge 2\) has a prime factor →
Product of odds is odd →
\((p\rightarrow q)\wedge p\) simplifies to →

(Expected style: induction; contrapositive; cases/parity; contradiction; contradiction; strong induction; direct; \(p\wedge q\).)

Mini repair log (use after scoring)

Item #	Failure mode	Day to reread	Retest (Y/N)
	wrong \(\rightarrow\) expansion	23–25
	quantifier negation incomplete	26–27
	missing witness in direct proof	28
	lowest terms omitted in \(\sqrt{2}\)	29
	incomplete cases	30
	vague \(P(n)\) / horses	31–32
	fallacy without counter-model	33

More mixed warm-ups (post-exam or second pass)

W1. Negate \(\forall\varepsilon>0\,\exists N\,\forall n\ge N\,(|a_n-L|<\varepsilon)\) fully.
W2. Prove: if \(a\mid b\) and \(a\mid c\) then \(a\mid(bx+cy)\) for all integers \(x,y\).
W3. Cases: \(n^2\equiv 0\) or \(1\pmod 4\).
W4. Induction: \(\sum_{i=0}^n 2^i = 2^{n+1}-1\).
W5. Fallacy: from “if sorted then binary search OK” and “binary search returned index” infer “sorted.”

Exit criteria (restate)

You may enter Stage IV when, without notes, you can: finish a small truth table; negate a two-quantifier sentence; write direct + contrapositive + induction proofs to peer standard; name and refute three fallacies. Honesty over speed—retest beats pretend-ready.

Tomorrow

Day 35 — Sets. Roster and builder notation; \(\in\), \(\subseteq\), \(\subsetneq\); empty set; universe; Russell paradox awareness; equality by double inclusion.