Mathbook: Introduction to Sets

Quite a while ago, an endeavoring individual tried to start an open-source repository of mathematical information called Mathbook. I contributed an article, which I’ll put down in two parts on this blog.

It seems that the project has died, although the website is still available. While this is a bit of a shame, I would like to give some of my own little lessons here. The creator’s idea behind Mathbook was to focus on giving people an understanding of why we do math in a certain way. This is missing from mathematical curriculum today, but it is vital to understand that when math was developed, decisions were made for specific reasons. Moving forward, I’ll occasionally add a new post here to that effect. The people in my life don’t always understand the math I learned, so this is part of my effort in showing it.

Introduction to Sets

We will learn the basics of how sets are used in mathematics. It is important to understand basic arithmetic before diving in, but nothing else.

 

Understanding Sets

In any field of mathematics, it is important to be able to deal with objects and structures. At the lowest level of mathematical objects and structures are sets. Most simply, a set is a collection of objects. We can think of the set of all flowers in Hawaii, or the set of whole numbers between 10 and 37. Typically, we use curly brackets (braces) to denote a set, such as \{1, 2, 3\} . If we are using the same set many times in a row, or talking about a set that cannot easily be written down, we can use some other symbol. Throughout this tutorial we will let S be the set \{1, 2, 3\}, and H be the set of all flowers in Hawaii.

 

There are certain rules and terms used with sets that allow mathematicians to be consistent when using and talking about sets. For example, we want to know what to call the objects in our sets in general, and how we can write sets.


Definition

Element: Each object or member of a set is called an element of the set. Each element can only occur once in a set. \{1,2,1,3\} is not a valid set since the 1 occurs twice. In addition, the order of elements in a set does not matter. S = \{1,2,3\} = \{3,1,2\}.

To say an element s is in the set S, we write s \in S.


It is also natural to discuss how many elements are in a set.


Definition

Cardinality: The number of elements in a set is called the cardinality of the set. The cardinality is often denoted by putting vertical bars around the set. For example, since S has three elements, we write |S| = 3.


For this tutorial, we will only be looking at sets with finite cardinality; this means we will always be able to list and count every element in the set. Future tutorials may explore larger sets, which becomes an even more powerful (and fun!) mathematical tool.

Often we want to look at some of the elements in a set, but not all of them. For example, we might want the elements of H which are red flowers. This is a very common pattern in mathematics: given an object or structure, how can we look at smaller objects that have a similar structure?


Definition

Subset: If every element in some set A is in a set B, we say that A is a subset of B, and we write A\subset B. For example, the set \{1,3\} is a subset of S.


When doing mathematics, it is good practice to look at the simplest example of any object you are interested in exploring. When it comes to sets, it becomes natural to ask “What if my set has no elements?”


Definition

Empty Set: The empty set is defined to be the set which has no elements. The most common notation is \emptyset, but you may see \{\}, especially in older math texts. This second notation emphasizes it is a set with no elements.


Almost everything you see and do has sets hiding in the background. They are a universal way of communicating mathematical ideas, and are thus very important to understand.

In a future post, I’ll give the second half of this post: Understanding Functions. Once you have established your structure (the set), and have explored some basic ideas (subsets, the empty set), it is important to discuss how you can have two sets interact. The simplest way to have two sets interact is via a function.

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