Set Theory Summary

Discrete Math · Axiom Academy

Let's review how sets provide the fundamental framework for organizing mathematical objects and underpin databases, probability, and abstract algebra. Definition: A set is a well-defined collection of distinct objects called elements or members Membership: We write a ∈ A to mean "a is an element of set A" and a ∉ A for "a is not in A" Roster Notation: Lists all elements explicitly, like A = 1, 2, 3, 4 Set-Builder Notation: Describes elements by property, like B = x | x > 0 meaning "all x such that x is positive" Common Sets: ℕ (natural numbers), ℤ (integers), ℚ (rationals), ℝ (real numbers) Subset: A ⊆ B means every element of A is also in B. We read this as "A is a subset of B" Proper Subset: A ⊂ B means A ⊆ B but A ≠ B (A is missing at least one element of B) Set Equality: A = B if and only if A ⊆ B and B ⊆ A (they contain exactly the same elements) Empty Set: The empty set ∅ = contains no elements and is a subset of every set Universal Set: U represents the universe of discourse containing all objects under consideration Union: A ∪ B = x | x ∈ A or x ∈ B – elements in at least one set Intersection: A ∩ B = x | x ∈ A and x ∈ B – elements in both sets Difference: A − B = x | x ∈ A and x ∉ B – elements in A but not in B Complement: A' = x | x ∈ U and x ∉ A – all elements not in A (relative to universal set U) Disjoint Sets: Sets A and B are disjoint if A ∩ B = ∅ Visual Representation: Venn diagrams use overlapping circles to represent sets and their relationships

This is the written version of the interactive lesson above. See the full Discrete Math course.