Sets, Functions, and Relations

Abstract Algebra · Axiom Academy

LESSON Sets, Functions, and Relations A comprehensive exploration of set operations, function properties, and equivalence relations—the foundational building blocks of abstract algebra. Sets are collections of distinct objects. The fundamental operations on sets are union (∪), intersection (∩), and difference (∖). Union: A ∪ B = x : x ∈ A or x ∈ B Intersection: A ∩ B = x : x ∈ A and x ∈ B Difference: A ∖ B = x : x ∈ A and x ∉ B A function f : A → B is a rule that assigns to each element in the domain A exactly one element in the codomain B. The range (or image) is the set of all actual outputs. Domain: The set of all possible inputs (A) Codomain: The set where outputs live (B) Range: The set of actual outputs, Im(f) = f(x) : x ∈ A ⊆ B 3. Injective (One-to-One) Functions A function f : A → B is injective if different inputs always produce different outputs. Formally: if f(x₁) = f(x₂), then x₁ = x₂. 4. Surjective (Onto) Functions A function f : A → B is surjective if every element in the codomain B is the image of at least one element in the domain A. The range equals the codomain: Im(f) = B. A function f : A → B is bijective if it is both injective and surjective. This means there is a perfect one-to-one correspondence between elements of A and elements of B. An equivalence relation on a set A is a binary relation ~ that is reflexive, symmetric, and transitive. It partitions A into disjoint equivalence classes . Reflexive: a ~ a for all a ∈ A Symmetric: If a ~ b, then b ~ a

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