majiayu000 CS Foundations Skill
Skill Metadata
skill_config:
version: "1.0.0"
category: theoretical
prerequisites: []
estimated_time: "6-8 weeks"
difficulty: intermediate
parameter_validation:
topic:
type: string
enum: [logic, proofs, sets, functions, combinatorics, number-theory, graphs]
required: true
depth:
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enum: [intro, standard, advanced]
default: standard
retry_config:
max_attempts: 3
backoff_strategy: exponential
initial_delay_ms: 500
observability:
log_level: INFO
metrics: [topic_usage, proof_verification_rate, exercise_completion]
Quick Start
Computer science is built on mathematics. Master these fundamentals:
Core Topics
Discrete Mathematics
- Set theory and operations
- Logic and proof techniques
- Combinatorics and counting
- Number theory basics
- Relations and functions
Computational Thinking
- Problem decomposition
- Abstraction and generalization
- Pattern recognition
- Algorithmic thinking
Formal Logic
- Propositional logic
- Predicate logic
- Proof by induction
- Truth tables and logical equivalence
Learning Path
Week 1: Logic Basics
- Boolean algebra
- Truth tables
- Logical operators
- Inference rules
Week 2: Proof Techniques
- Direct proof
- Proof by contradiction
- Mathematical induction
- Strong induction
Week 3: Set Theory
- Set operations (∪, ∩, complement)
- Cartesian product
- Relations
- Equivalence relations
Week 4: Functions
- Function notation
- Domain, codomain, range
- One-to-one and onto
- Function composition
Week 5: Combinatorics
- Counting principles
- Permutations
- Combinations
- Pigeonhole principle
Week 6: Number Theory
- Modular arithmetic
- Prime numbers
- GCD and Euclidean algorithm
- Congruence
Practice Problems
- Prove by induction that 1+2+...+n = n(n+1)/2
- Prove √2 is irrational
- Show A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- Count functions from {1,2,3} to {a,b}
- Solve: x ≡ 5 (mod 12) and x ≡ 3 (mod 8)
Troubleshooting
| Issue | Root Cause | Resolution |
|---|---|---|
| Proof stuck | Missing case or wrong direction | Check base case, verify induction step |
| Set operation confusion | ∪ vs ∩ mix-up | Draw Venn diagram |
| Counting error | Overcounting duplicates | Distinguish P(n,r) vs C(n,r) |
| Modular arithmetic error | Forgot wraparound | Work with remainders explicitly |
Key Concepts
- Axioms: Statements we assume true
- Theorems: Statements we prove
- Lemmas: Helper theorems
- Corollaries: Results that follow easily
Why It Matters
These foundations enable:
- Understanding algorithm correctness
- Analyzing computational complexity
- Designing new algorithms
- Proving algorithm properties
- Understanding what's computable
Interview Prep
- Explain mathematical induction
- Prove that a function is injective
- Count permutations with constraints
- Solve modular equations
- Apply pigeonhole principle