Set theory

Abstract

We shall assume the reader to be familiar with the elements of set theory. Nonetheless, we begin with a review of certain set-theoretic fundamentals, largely to fix notation and terminology. (Readers wishing to improve their acquaintance with set theory, or to pursue in greater depth any of the topics touched on below, might consult [31] or [34]; another excellent source for most topics is [10].) For one thing, at the most elementary level, we reserve certain symbols throughout the book for several important sets. The system of positive integers is denoted by ℕ, the system of nonnegative integers by ℕ₀ the system of all integers by ℤ, the real number system by ℝ, and the complex number system by ℂ. The empty set is denoted by ∅, and if X and Y are any two sets, the set-theoretic difference {x ∈ X: x ∉ Y} is denoted by X\Y and the symmetric difference (X\Y) ∪ (Y\X) by X ∇ Y. Moreover, if f is a mapping of X into Y (notation: f:X → Y) and A ⊂ X and B ⊂ Y, then f(A) will denote the set {f(x): x ∈ A} and f⁻¹(B) the set {x ∈ X: f(x) ∈ B}.