6120a Discrete Mathematics And Proof For Computer Science Fix Today
Report: 6.120A Discrete Mathematics and Proof for Computer Science
5. Pedagogical Approach
To ensure students grasp the "Fix" (rigorous nature) of the subject, the course employs:
- Active Learning: Students participate in "proof workouts" where they critique invalid proofs and construct valid ones.
- Integration with Code: Concepts are reinforced through programming assignments (e.g., implementing a recursive function and proving its correctness via induction).
- Automated Grading Tools: Use of systems like Gradescope or custom proof-checkers to provide immediate feedback on logic syntax.
Module A: Mathematical Logic and Proofs
- Propositional Logic: Operators, truth tables, normal forms (CNF/DNF).
- Predicate Logic: Quantifiers ($\forall, \exists$), nested quantifiers, and rules of inference.
- Proof Strategies:
- Direct Proof and Counterexamples.
- Proof by Contraposition and Contradiction.
- Induction: Mathematical Induction, Strong Induction, and Structural Induction (crucial for data structures like trees and lists).
Fix 5.3: Cycle Detection (for invariants)
To prove no odd cycle exists (bipartite graphs): Report: 6
- Fix: Assume a cycle
v1, v2, ..., vk, v1. Color vertices alternately. For an odd cycle, you would need the same color on adjacent vertices, a contradiction. Always use a parity argument.
B. Proof Techniques
Four main types cause trouble:
| Proof Type | Strategy | Typical Mistake | Fix |
|------------|----------|----------------|-----|
| Direct | Assume P, derive Q | Circular reasoning | Start with given facts, use definitions |
| Contrapositive | Prove ¬Q → ¬P | Confusing with contradiction | State contrapositive explicitly |
| Contradiction | Assume P ∧ ¬Q, reach impossible | Not reaching a clear contradiction | End with “this contradicts X” |
| Induction | Base case + inductive step | Forgetting base case or assuming what you’re proving | Write inductive hypothesis clearly | Module A: Mathematical Logic and Proofs
Fix for induction: Always show P(k) → P(k+1) without assuming P(k+1). Fix 3.2: Function Injectivity/Surjectivity
Write-Up: Mastering 6120A – Discrete Mathematics and Proof for Computer Science
2.6 Graph Theory
- Undirected/directed graphs, vertices, edges, degree.
- Paths, cycles, connectivity, components.
- Trees: properties, rooted trees, spanning trees.
- Eulerian and Hamiltonian paths.
- Graph coloring, bipartite graphs.
- Applications: social networks, routing, dependency resolution, scheduling.
Fix 3.2: Function Injectivity/Surjectivity
- Injective (one-to-one):
f(a) = f(b) ⇒ a = b. Fix: Assume f(a)=f(b), then algebraically cancel to get a=b.
- Surjective (onto): For every
y in codomain, exists x with f(x)=y. Fix: Let y be arbitrary. Solve f(x)=y for x in terms of y. Verify x is in domain.
Common 6120a exam trick: Prove f is bijective by doing both.