Naive Set Theory Summary

Set Theory · Axiom Academy

Key concepts, notation, and operations for working with sets Roster notation: List elements explicitly like 1, 2, 3 Set-builder notation: Describe with conditions like x : x > 0 Membership: means x is in A; means x is not in A Subsets: means every element of A is also in B Empty set: contains no elements and is a subset of every set Union : Elements in A or B (combines both) Intersection : Elements in A and B (common elements) Difference : Elements in A but not in B Complement : Elements in U but not in A Symmetric difference: In exactly one of A or B, not both Example: Solving a Set Problem Identify the sets: Write out A, B, and any universal set U explicitly Identify the operation: Determine if you need union, intersection, difference, or complement Apply the definition: Go element by element checking membership conditions Verify with inclusion-exclusion: Check that |A ∪ B| = |A| + |B| − |A ∩ B| holds Use Venn diagrams: Visualize overlapping regions to confirm your answer Commutative: Order doesn't matter for ∪ and ∩ Associative: Grouping doesn't matter for ∪ and ∩ Distributive: Both directions work (unlike numbers!) Idempotent: A ∪ A = A and A ∩ A = A Cartesian product: |A × B| = |A| · |B| ordered pairs Not commutative: Set difference and Cartesian product depend on order

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