A.1 Sets and maps

A setX is a collection of objects, which we call elements. We write x ∈ X if x is an element of X. We also write X = {x, y,…} to denote the elements of X. The empty set ∅ = {} is the unique set containing no elements. A set U is a subset of X, written U ⊆ X, if x ∈ U implies x ∈ X. A set U is a proper subset of X, written U ⊂ X, if U ⊆ X and U ≠ X. The unionX ∪ Y of two sets X and Y is the set of all elements in X or in Y (or in both), whereas the intersectionX ∩ Y is the set of all elements that are in both X and Y. If X ∩ Y = ∅ then XmeetsY (and Y meets X). The (set-theoretic) difference of two sets X and Y is the set X\Y = X − Y = {x ∈ X : x ∉ Y}. (This definition does not require that Y be contained in X.) The complement of U ⊆ X is Ū ≔ X − U.