cs-foundations

Research

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

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Source SKILL.md: https://github.com/foryourhealth111-pixel/Vibe-Skills/blob/HEAD/bundled/skills/cs-foundations/SKILL.md

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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

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