Introduction to Set Theory

Interactive Lesson by Thomas Cordell

Advanced Sets
Text Box: In this section we will cover the Cartesian product of sets, power sets, and cardinality. Cartesian Product The Cartesian product are the sets of all pairs between the two sets. Let Q and R be sets. By Q x R we mean the set of all “pairs” (q, r) such that q is an element in Q and r an element in R. This can be dictated in symbols, such that: Q x R := {(q, r) | q Î Q, r Î R} The set Q x R is called the Cartesian product of Q and R. For example, setting Q := {5, 6, 7} And setting R := {10, 20} Then we have Q x R = {(5, 10), (6, 10), (7, 10), (5, 20), (6, 20), (7, 20)} And R x Q = {(10, 5), (10, 6), (10, 7), (20, 5), (20, 6), (20, 7)} Power Sets A power set is the set of all subsets of the set. The power set is denoted by P(set name). Important Note: Every set contains an empty set as a subset. Always remember order of sets is irrelevant. {1, 2} is the same as {2, 1} and you only need to list the set once. Also remember that the set itself is a subset of the set. For example, setting V := {1, 2, 3} The P(V), power set of V, is: P(V) = { Ø, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} Cardinality The cardinality of a set is a measure of the number of elements the set contains. The cardinality is denoted by placing a bar on both sides of the set. For example, | A |, means the cardinality of set A. This tool is very useful for comparing sets. For example, setting A := {1, 2, 3) and B := {4, 5, 6} We have | A | = 3 and | B | = 3 And can say | A | = | B | Important Note: Be mindful of what is being compared. The statement | A | = | B | does not mean that the sets are the same, it means that both sets have the same number of elements. Conclusion: You have completed all required sections on Set Theory Lesson. You should now be familiar with the basics of sets and the many different functions used to create, describe, and compare sets. You are now ready to complete the Set Theory Assignment.