cs-foundations

CS Foundations Skill

Skill Metadata

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  version: "1.0.0"
  category: theoretical
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  estimated_time: "6-8 weeks"
  difficulty: intermediate

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      enum: [logic, proofs, sets, functions, combinatorics, number-theory, graphs]
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    depth:
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      enum: [intro, standard, advanced]
      default: standard

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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

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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

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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)

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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 |

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Key Concepts

• Axioms: Statements we assume true
• Theorems: Statements we prove
• Lemmas: Helper theorems
• Corollaries: Results that follow easily

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Why It Matters

These foundations enable:

• Understanding algorithm correctness
• Analyzing computational complexity
• Designing new algorithms
• Proving algorithm properties
• Understanding what's computable

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Interview Prep

• Explain mathematical induction
• Prove that a function is injective
• Count permutations with constraints
• Solve modular equations
• Apply pigeonhole principle