The Joy of Sets

A colony of beavers, an unkindness of ravens, a murder of crows, a team of oxen, . . . each is an example of a set of things. Rather than define what a set is, we assume you have the “ordinary, human, intuitive (and frequently erroneous) understanding”1 of what a set is. 1 Paul Halmos, Naive Set Theory, Springer–Verlag, NY 1974. Sets have elements, often called members. The elements of a set may be flies, beavers, words, numbers, sets, vectors, . . . If x is some object and S is a set, we write x ∈ S if x is an element of S and x 6∈ S if x is not a member of S. For us, the most important property a set S has is this: if x is an object, then either x ∈ S or x 6∈ S, but not both. The sets N, Z, Q, and R are well-known to you, though you may not know their names. The set of natural numbers is denoted by N, Math 582, Introduction to Set Theory, provides a rigorous treatment of N. and its elements are the numbers 1, 2, 3, 4, . . . Note that if n, m ∈ N, then n + m ∈ N; that is, N is closed under addition. However, N is not closed under subtraction. For example, (4− 5) 6∈ N. To overcome this inconvenience we consider Z, the set of integers, which has as its The symbol Z is derived from Zahlen, the German word for numbers. elements the numbers 0, ±1, ±2, ±3, . . . While Z is closed under addition, subtraction, and multiplication, it is not closed under division. never divide by zero. For example, (−23)/57 6∈ Z. To surmount this difficulty, we form the set of rational numbers, Q. Intuitively, Q is the set of all numbers that In Math 412, Introduction to Modern Algebra, Q is rigorously defined. can be expressed as a fraction n/m with n ∈ Z and m ∈ N. While closed under multiplication, division, addition, and subtraction, Q is missing important numbers like √ 2. There are many ways to overApproximately 1.41, √ 2 is the ratio of a square’s diagonal to one of its sides. It was probably the first known example of a number that does not belong to Q. come this inconvenience; the most common approach is to introduce R, the set of real numbers. R is usually depicted as a line that extends Introductory analysis courses, including Math 351 and Math 451, provide indepth treatments of R. forever in both directions.