Introduction to Set Theory

Interactive Lesson by Thomas Cordell

Sets In Action
Text Box: Now that you have the basic understanding of sets, lets introduce a few more symbols and some associated terminology related to how sets interact. Subsets A subsets is a set that is contained inside another set. It’s a relationship between sets. A subset is notated with the following symbol: Í For example: {12, 22} Í {3, 12, 15, 22} Important note: Two sets R and S are called equal if R Í S and S Í R. This this case, we write R = S. For example, assume that R has no element and S has no element. Then R = S. This means there is really only one set without any elements. This is denoted as Ø and is called an “empty set”. Intersections An intersection of two sets X and Y is the set that contains all elements of X that also belong to Y. The intersection is denoted using symbol: Ç For example, setting X: = {1, 2, 3, 4, 5} And setting Y: {4, 5, 6, 7, 8} We have that X Ç Y = {4, 5} Unions A union of two sets forms a single set of distinct elements of both sets. The union of two sets is denoted using symbol: È For example, setting X: = {1, 2, 3, 4, 5} And setting Y: = {4, 5, 6, 7, 8} We have that X È Y = {1, 2, 3, 4, 5, 6, 7, 8} Symmetric Difference The symmetric difference of two sets is the set of elements which are in one of the sets, but not in both. Symmetric difference is denoted using symbol: \ For example, setting X: = {1, 2, 3, 4} And setting Y: = {3, 4, 7} We have that X \ Y = {1, 2} Important Note: In general, X \ Y and Y \ X are not the same. In the example above X \ Y = {1, 2} and Y \ X = {7} Conclusion: You have now learned the concepts of subsets, intersections, unions, and symmetric difference. These are all very useful methods to create new sets from given ones. We will move on now to cover the more advanced functions of sets.