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“6120A: Discrete Mathematics and Proof for Computer Science”
(with a focus on “fix” — likely meaning a corrected, revised, or definitive syllabus / topic guide)
This write-up is designed as a comprehensive reference for instructors or advanced students, covering motivation, core topics, proof techniques, and computational connections. This write-up is designed as a comprehensive reference
4. Sample Problems (Typical for 6.120A)
- Logic: Show that
(P → Q) ∧ (Q → R) logically implies (P → R).
- Induction: Prove that for all n ≥ 1, sum_i=1^n i = n(n+1)/2.
- Combinatorics: How many binary strings of length 10 have exactly three 1’s? (Answer: C(10,3)=120)
- Number Theory: Compute 7^120 mod 13 using Fermat’s theorem.
- Graphs: Prove that any connected graph with n vertices has at least n-1 edges.
- Relations: Show that congruence modulo m is an equivalence relation.
5. Why Discrete Math is Essential for Computer Science
| Area of CS | Discrete Math Concept Used |
|------------|----------------------------|
| Algorithms | Induction, recurrences, invariants |
| Data structures | Trees, graphs, sets, functions |
| Complexity theory | Counting, pigeonhole principle |
| Cryptography | Modular arithmetic, primes |
| Compilers | Finite automata, regular languages |
| Databases | Relational algebra (sets, functions) |
| Machine learning | Combinatorics (permutations for feature selection) |
| Software verification | Logic, proofs of correctness | Claim : ∀n ∈ ℕ
Without discrete math, a computer scientist cannot: Example fixed template (Induction):
- Argue that an algorithm terminates or is correct.
- Analyze worst-case time/space complexity rigorously.
- Design or break cryptographic protocols.
- Understand computability or formal language hierarchies.
E. Graph Theory
- Degree, paths, cycles, trees – basic definitions matter.
- Fix: Memorize: sum of degrees = 2×edges. A tree on (n) vertices has (n-1) edges.
- Proof fix for graphs: Use handshaking lemma for degree-related proofs.
2.3 Proof Methods (Fixed Canon)
Each proof must be prefaced by proof strategy label:
- Direct proof – assume premise, derive conclusion.
- Proof by contrapositive – prove ¬Q → ¬P instead of P → Q.
- Proof by contradiction – assume P ∧ ¬Q, derive contradiction.
- Proof by cases – exhaustive case split.
- Mathematical induction – base case + inductive step (simple, strong, well‑ordering).
- Structural induction – for recursively defined sets (strings, trees).
Example fixed template (Induction):
Claim: ∀n ∈ ℕ, n ≥ 1 → P(n)
Proof (by simple induction on n):
Base case n = 1: …
Inductive hypothesis: Assume P(k) for some arbitrary k ≥ 1.
Inductive step: Show P(k+1) using the hypothesis.
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