### Unit 1: Sets, Set Relations, and Set Functions

Computer scientists often find themselves working with groups of homogeneous or heterogeneous entities. Mathematicians devised single-membership set theory to respond to these situations. In this unit, we will cover the theoretical background of sets and take a look at associated definitions, notations, relations, and functions. This is a fundamental tool of mathematics and computer science, and is essential to understanding the other topics in this course.

**Completing this unit should take you approximately 5 hours.**

Upon successful completion of this unit, you will be able to:

- define sets, operations on sets, and state important set properties;
- categorize sets into their various types (such as singleton, finite, infinite, equal, null, proper subset, and improper subset) and give a definition of each;
- describe set notation and how that notation is used to perform operations via symbol manipulation; and
- apply set definitions, operations, and properties to demonstrate set membership within a specific context.

### 1.1: Set Notation and Relations

Every field of study seeks a common terminology and symbology. While it is possible to think about a subject without knowing its particular language, it is not possible to communicate with others about that subject without some common frame of reference. Thus we begin with the basic terms and notations of set theory.

This table defines the notation used in this course.

Work these exercises to see how well you understand this material. Hints and solutions to some of these exercises are given on the second page. As the course proceeds, new notations and terminology will be introduced as they are needed. Be sure to not plug-and-chug answers to the exercises. These are to aid your understanding, and it will be difficult to pass the exam if you do not take the exercises seriously. While you should not copy others' work, feel free to discuss these exercises with others who are taking this course.

### 1.2: Basic Set Operations

Even as set members are discrete, so are sets themselves. The question we ask about each member is, "Of what sets is it entirely a member?" Although there are no partial set memberships, an entity can be entirely a member of more than one set. So, we can perform various operations on sets, such as add one set to another and subtract one set from another. With questions that require a Yes or No response, there is no dual membership since those are on the same hierarchical level. However, with layered hierarchies, dual 100% membership is possible. For example, we can talk about a car that is also a vehicle. The entity exists entirely in two different sets. Later in the course, we will talk more about hierarchies.

Read this page to become familiar with the various types of sets. This page is an aid to set terminology and notation.

Work these exercises to see how well you understand this material.

### 1.3: Cartesian Products and Power Sets

While the last section discussed combining sets of individual members to create another set of individual members, here we discuss creating sets of non-repeating tuples (pairs, triplets, and higher groupings). Later in the course, we will see how to calculate the number of tuples that would be created under various circumstances.

Work these exercises to see how well you understand this material.

### 1.4: Summation Notation and Generalizations

We have dealt with relatively straightforward notation so far. However, as situations involving sets become more complex, a more compact notation is needed. Here presented is an introduction to that notation.

Work these exercises to see how well you understand this material.

### Unit 1 Assessment

- Receive a grade
Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.

- This assessment