| Focus | Activities |
|---|
| Content Review | Read all lecture notes systematically |
| Gap Identification | List topics you find confusing |
| Resource Gathering | Collect textbook sections, homework solutions |
| Initial Organization | Create folder structure for study materials |
| Focus | Activities |
|---|
| Concept Mapping | Draw connections between topics visually |
| Definition List | Compile ALL definitions in one document |
| Theorem List | List all theorem statements (no proofs yet) |
| Weak Area Focus | Spend extra time on confusing topics |
| Flashcard Creation | Start making definition/theorem flashcards |
| Focus | Activities |
|---|
| Daily Proofs | Practice 3–5 proofs every day |
| Study Group | Form group, meet 2–3 times per week |
| Flashcard Review | Daily flashcard sessions (20–30 min) |
| Office Hours | Visit instructor/mentors with specific questions |
| Practice Exams | Attempt old TM questions under timed conditions |
| Focus | Activities |
|---|
| Review Session | Attend instructor’s review session |
| Practice Problems | Work through comprehensive problem sets |
| Definition Mastery | Can recite all definitions word-perfect? |
| Theorem Recall | Practice stating theorems precisely |
| Mock Exam | Take full 120-min practice test |
| Group Quizzing | Quiz each other on random topics |
| Focus | Activities |
|---|
| Light Review | Skim through notes (don’t cram new material) |
| Flashcard Final Pass | Review all flashcards one last time |
| Rest & Recovery | Reduce study intensity, avoid burnout |
| Confidence Building | Review what you DO know well |
| Focus | Activities |
|---|
| Minimal Study | 30–60 min light review only |
| Sleep Priority | Get 8 hours of quality sleep |
| Good Nutrition | Eat healthy meals, stay hydrated |
| Mental Prep | Visualize success, stay positive |
| Logistics | Know exam location, bring ID |
| Type | When to Use | Template | Example |
|---|
| Direct | Straightforward implication | Assume P. Show Q follows. | If \(n\) even, then \(n^2\) even |
| Contradiction | Proving impossibility | Assume \(\lnot Q\). Derive contradiction. | \(\sqrt{2}\) is irrational |
| Contrapositive | Negation easier than direct | Prove \(\lnot Q \to \lnot P\) instead of \(P \to Q\) | If \(n^2\) odd, then \(n\) odd |
| Induction | Properties of \(\mathbb{N}\) | Base case + inductive step | \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\) |
| Construction | Existence claims | Build explicit example | Bijection between \(A\) and \(B\) |
| Cases | Multiple scenarios | Case 1: …, Case 2: … | Prove for even and odd separately |
✍️ Structure Every Proof
- Opening: State what you’re proving
- Setup: Define variables, state assumptions
- Body: Logical argument with clear steps
- Conclusion: “Therefore, …” or “Thus, …” or \(\square\)
Bad: “It’s obvious that…”
Good: “By definition of X, we have… Therefore…”
| Category | Must Know |
|---|
| Set Laws | Commutative, associative, distributive, De Morgan’s, identity, complement, idempotent, domination, absorption, involution |
| Power Set | \(\mathcal{P}(A) = \{B \mid B \subseteq A\}\), \(|\mathcal{P}(A)| = 2^{|A|}\) |
| Cantor’s Theorem | \(|A| < |\mathcal{P}(A)|\) for any set \(A\) |
| Schroeder-Bernstein | If exists injection \(A \to B\) and exists injection \(B \to A\), then exists bijection \(A \leftrightarrow B\) |
| Category | Must Know |
|---|
| Properties | Reflexive: \(\forall x ~ (xRx)\); Symmetric: \(\forall x,y ~ (xRy \to yRx)\); Transitive: \(\forall x,y,z ~ (xRy \land yRz \to xRz)\); Antisymmetric: \(\forall x,y ~ (xRy \land yRx \to x=y)\) |
| Equivalence | Reflexive + symmetric + transitive \(\leftrightarrow\) partition |
| Functions | Injective: \(f(a)=f(b) \to a=b\); Surjective: \(\forall y ~ \exists x ~ f(x)=y\); Bijective: both |
| Composition | \((g \circ f)(x) = g(f(x))\); associative; inverse if bijective |
| Category | Must Know |
|---|
| Boolean Laws | Identity, Null, Idempotent, Complement, De Morgan’s, Absorption |
| Normal Forms | DNF: OR of ANDs; CNF: AND of ORs |
| Completeness | \(\{\land, \lor, \lnot\}\), \(\{\text{NAND}\}\), \(\{\text{NOR}\}\) are functionally complete |
| Gate Symbols | Know circuit symbols for AND, OR, NOT, NAND, NOR, XOR |
| Category | Must Know |
|---|
| Truth Tables | \(\lnot, \land, \lor, \to, \leftrightarrow\) truth values for all combinations |
| Natural Deduction | Modus Ponens (\(P, P\to Q \vdash Q\)), Modus Tollens (\(\lnot Q, P\to Q \vdash \lnot P\)), \(\land\)-Intro, \(\lor\)-Elim, etc. |
| Soundness vs Completeness | Soundness: \(\vdash\) implies \(\models\) (no false proofs); Completeness: \(\models\) implies \(\vdash\) (can prove all truths) |
| Quantifiers | \(\lnot(\forall x ~ P(x)) \equiv \exists x ~ \lnot P(x)\); \(\lnot(\exists x ~ P(x)) \equiv \forall x ~ \lnot P(x)\) |
- Active Recall: Test yourself without looking at notes – strengthens memory retrieval
- Spaced Repetition: Review material at increasing intervals to fight forgetting
- Teach Others: Explain concepts to study partners – best test of true understanding
- Practice Exams: Simulate exam conditions to build familiarity and reduce anxiety
- Whiteboard Practice: Work problems standing at a whiteboard – engages different thinking patterns
✅ Do This:
- Meet regularly (2–3 times per week)
- Quiz each other on definitions/theorems
- Work through proofs together
- Explain difficult concepts to each other
- Share different solution approaches
❌ Don’t Do This:
- Just socialize without studying
- Let one person do all the explaining
- Skip individual preparation before meeting
- Argue about minutiae; ask instructor to clarify
- Study only in groups (need solo time too!)
| Resource | Priority |
|---|
| Lecture Slides | ⭐⭐⭐⭐⭐ |
| Textbook | ⭐⭐⭐⭐ |
| Homework Solutions | ⭐⭐⭐⭐ |
| Review Sessions | ⭐⭐⭐⭐⭐ |
| Mentors/Office Hours | ⭐⭐⭐⭐⭐ |
- Online Videos: When reading isn’t clicking
- Math StackExchange: For alternative perspectives
- Practice Problems: Throughout preparation
- Old Exams: Final week preparation
| Mistake | Why It’s Wrong | Fix |
|---|
| Circular Reasoning | Assumes what you’re proving | Ensure logical flow: premises \(\to\) conclusion |
| Using “Obvious” | Skips justification | Provide explicit reasoning |
| Confusing Necessary/Sufficient | \(P\to Q\): \(Q\) necessary for \(P\), \(P\) sufficient for \(Q\) | Remember the direction! |
| Wrong Quantifier Order | \(\forall x ~ \exists y ~ P(x,y) \neq \exists y ~ \forall x ~ P(x,y)\) | Be precise with quantifier scope |
| Incomplete Cases | Miss edge cases | Systematically check all scenarios |
| Mistake | Example | Correction |
|---|
| No setup | “Therefore \(x = 5\)” | “Let \(x\) be arbitrary. Then…” |
| Jumping steps | “Clearly \(A = B\)” | Show intermediate steps |
| Poor notation | Using same variable for different things | Define all variables clearly |
| No conclusion | Proof just stops | End with “Therefore…” or \(\square\) |
✅ Light review (1 hour max):
- Skim definition flashcards
- Glance at theorem list
- Review 1–2 key proofs
✅ Self-care:
- 8 hours of sleep (non-negotiable!)
- Healthy meals throughout the day
- Light exercise or walk
- Relaxation techniques (deep breathing, meditation)
✅ Logistics:
- Confirm exam time and location
- Prepare ID and any allowed materials
- Set multiple alarms
- Plan to arrive 10 min early
- ❌ Don’t cram new material – if you don’t know it now, won’t learn it tonight
- ❌ Don’t stay up late studying – sleep > cramming
- ❌ Don’t drink excessive caffeine – disrupts sleep quality
- ❌ Don’t panic – trust your preparation
- ❌ Don’t compare yourself to others – focus on your own readiness
🎯 Positive Self-Talk
- “I’ve prepared thoroughly for this.”
- “I know the material well.”
- “Partial credit means every bit of knowledge counts.”
- “I can handle difficult questions – I’ll do my best.”
- “One exam doesn’t define me or my understanding.”
| Situation | Response |
|---|
| Can’t remember definition | Move on, come back later; write related concepts |
| Proof seems impossible | Write setup, state approach, show what you can |
| Running out of time | Prioritize high-value questions; outline remaining proofs |
| Feeling anxious | Deep breath, 30-second break, refocus |
| Blank mind | Read question again slowly; write anything related |
Remember: TMs test understanding, not just memorization. Focus on:
- Why definitions are structured that way
- How theorems connect to each other
- When to apply different proof techniques
- What the big ideas are, not just details
Trust your preparation, stay calm, and do your best. Good luck! 🍀