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This assures us that finite cardinality behaves as one would expect. If one set A has cardinality n then it cannot also have cardinality m 6= n where m,n ∈ N. For supose that it did. Renaming m and n if necessary, we may suppose m > n. Then we would have two bijections f : A → In and g : A → Im. Then f g−1: Im → In is number of things in an in nite set, we will use the idea of bijection to determine when things are the same" size. Before we start developing theorems, let's get some examples working with the de nition of nite sets. Example 1. Fix m 2N. Let X m = fq 2Q j0 q 1; and mq 2Zg. Prove that X is nite, and determine its cardinality. Solution. To prove that X If you know the cardinality of sets, then you can compare them by size and determine which set is bigger. If the cardinality of two sets is the same, then there is a bijection between them. The input set can be specified in the standard set format, using curly brace characters { } on the sides and a comma as the element separator (for example {1, 2, 3}) and in a non-standard set format (for So the cardinality of B is three. The empty set, like any other set, can be put into another set. So {∅} is a set containing the empty set. Think of it as a box containing an empty box. The set {∅,{3,4}} has two elements: the empty set and the set {3,4}. 2 Powersets and set-valued functions The cardinality of a set is also called its size, when no confusion with other notions of size [2] is possible. The cardinality of a set is usually denoted , with a vertical bar on each side; [3] this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set may alternatively be denoted by , , , or . Set Notation Sameer Kailasa , Christopher Williams , Dylan Hendrickson , and 2 others contributed The cardinality of a set is a measure of a set's size, meaning the number of elements in the set. For instance, the set A = {1,2,4} A = {1,2,4} has a cardinality of 3 3 for the three elements that are in it. Introduction to Set Theory • A set is a new type of structure, represen2ng an unordered collec2on (group, plurality) of zero or more dis+nct (different) objects. Cardinality (i.e., size) of a set • If there are n dis2nct elements in the set S where n is a nonnegave integer, we say SET THEORY 3 6. Cardinality 000D The cardinalityof a set Ais the least ordinal size(S) = max ℵ 0,κ 1,κ 2), The Cantor set is a famous set of real numbers named after the mathematician Georg Cantor. We denote the Cantor set by Cand construct it in the following way: Exercise: Draw a picture for each step below to visualize the Cantor set. 1. Start with the closed interval [0;1]. 2. Remove the middle third of this interval 1 3;2 3. You should be left with 0;1 3 [2 3;1. 3. Proposition 2 is very useful since it allows one to compute the size of one set based on another set whose size is easy to compute. Let us apply this to determine the cardinality of power sets. Proposition 3. For any finite set A, jP(A)j= 2jAj. Proof. Consider an arbitrary finite set A. Let A = fa1, a2,. . . ang, i.e., jAj= n. Exercise 1.1. If Xis a nonempty set, then jXj jYjif and only if there is a sur