3: Sets
- Page ID
- 59937
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- 3.1: Basics
- An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set. A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}. Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a subset of the larger set of all Madonna albums.
- 3.2: Union, Intersection, and Complement
- Commonly sets interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets. The union of two sets contains all the elements contained in either set (or both sets). The intersection of two sets contains only the elements that are in both sets. The complement of a set A contains everything that is not in the set A .
- 3.3: Venn Diagrams
- To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations now called Venn Diagrams. A Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets. Basic Venn diagrams can illustrate the interaction of two or three sets.
- 3.4: Cardinality
- Often times we are interested in the number of items in a set or subset. This is called the cardinality of the set. The number of elements in a set is the cardinality of that set. The cardinality of the set A is often notated as |A| or n (A).
- 3.5: Exercises
- This page contains 46 exercise problems related to the material from Chapter 3.