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cs-foundations

Master discrete mathematics, logic, formal proofs, and computational thinking. Build the mathematical foundation for all computer science.

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  • SKILL.md3.3 KB
  • assets/proof-template.md1.4 KB
  • references/PROOF-TECHNIQUES.md3.4 KB
  • scripts/proof_validator.py2.2 KB
  • scripts/verify-proof.py5.0 KB

SKILL.md(原文)

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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:
      type: string
      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

  1. Prove by induction that 1+2+...+n = n(n+1)/2
  2. Prove √2 is irrational
  3. Show A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
  4. Count functions from {1,2,3} to {a,b}
  5. Solve: x ≡ 5 (mod 12) and x ≡ 3 (mod 8)

Troubleshooting

IssueRoot CauseResolution
Proof stuckMissing case or wrong directionCheck base case, verify induction step
Set operation confusion∪ vs ∩ mix-upDraw Venn diagram
Counting errorOvercounting duplicatesDistinguish P(n,r) vs C(n,r)
Modular arithmetic errorForgot wraparoundWork 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

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