A cheat sheet for mathematical objects and structures, useful for quick reference. Foundations of Sets and Relations Set A collection of distinct objects, often written as X = { x ∣ P ( x ) } X = \{x \mid P(x)\} X = { x ∣ P ( x )} , where P ( x ) P(x) P ( x ) is a property. No additional structure. Map / Function A mapping f : X → Y f: X \to Y f : X → Y assigns each x ∈ X x \in X x ∈ X to a unique y ∈ Y y \in Y y ∈ Y . Relation A subset R ⊆ X × X R \subseteq X \times X R ⊆ X × X , such as equivalence relations or partial orders. Basic Algebraic Structures Semigroup A set S S S with a binary operation ⋅ : S × S → S \cdot: S \times S \to S ⋅ : S × S → S such that: Associativity: For all a , b , c ∈ S a, b, c \in S a , b , c ∈ S , ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) (a \cdot b) \cdot c = a \cdot (b \cdot c) ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) . Monoid A semigroup with an identity element e ∈ M e \in M e ∈ M , satisfying: Identity: a ⋅ e = e ⋅ a = a ...
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