Problem Solving Guide

Set Theory · Axiom Academy

Master the strategies and techniques for solving set theory problems effectively. Element Chasing: For set equality, show x ∈ A implies x ∈ B and vice versa. Track individual elements through logical steps. Regular Induction: Prove base case (usually n = 0 or 1), then prove if true for n , it holds for n + 1 . Essential for finite set proofs. Transfinite Induction: For ordinal properties, prove for 0, show successor case, and prove limit ordinal case (take supremum of earlier cases). Diagonal Arguments: To prove uncountability or construct counterexamples, assume a listing and derive a contradiction by diagonalization. General Problem-Solving Strategy Understand the Goal: Identify what you're proving—existence, uniqueness, equality, bijection, or cardinality relationship. Identify Given Information: List all axioms, definitions, and hypotheses that apply. Note any special set properties. Choose the Right Approach: Direct proof, contradiction, contrapositive, or construction. Match technique to problem type. Work Both Directions: For "if and only if" statements, prove both implications separately and clearly. State the Goal: Show A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) . This requires proving both subset inclusions. First Direction (⊆): Let x ∈ A ∩ (B ∪ C) . Then x ∈ A and x ∈ B ∪ C . Case 1: if x ∈ B , then x ∈ A ∩ B . Case 2: if x ∈ C , then x ∈ A ∩ C . Either way, x ∈ (A ∩ B) ∪ (A ∩ C) .

This is the written version of the interactive lesson above. See the full Set Theory course.