Table of Contents
In this chapter, we focus on a basic idea that appears throughout all of mathematics: sets and the elements that belong to them.
A set is a collection of distinct objects, considered as a whole. The objects in a set are called its elements (or members). In simple language, a set is like a box, and the elements are the things inside the box.
We will not yet worry about advanced notation or diagrams; those will come later in the chapters on set notation and Venn diagrams. Here we build intuition for what sets and elements are and how to think about them.
Examples of sets and elements
You already use the idea of sets in everyday life, often without naming it.
- A set of days of the weekend: {Saturday, Sunday}
- Here the elements are “Saturday” and “Sunday”.
- A set of vowels in the English alphabet: {a, e, i, o, u}
- The elements are the letters a, e, i, o, u.
- A set of your friends: {Alex, Fatima, Diego}
- The elements are the people Alex, Fatima, Diego.
- A set of small whole numbers: {0, 1, 2, 3, 4}
- The elements are the numbers 0, 1, 2, 3, 4.
You can make sets out of many kinds of things:
- Numbers
- Letters
- People
- Shapes
- Even other sets (later you will see sets whose elements are themselves sets)
In all these cases, we are grouping objects together and treating the group as a single thing.
Membership: “being in” a set
The key idea for sets is membership: whether something is or is not an element of a given set.
If we have a set of vowels:
$$
V = \{a, e, i, o, u\},
$$
then:
- The letter
ais an element of the set $V$. - The letter
bis not an element of the set $V$.
When we say “$x$ is an element of the set $A$”, we are talking about membership: $x$ belongs to the group $A$.
It is always a yes-or-no question:
- Either something is in the set (it is an element),
- Or it is not in the set (it is not an element).
There is no “halfway” membership in this basic setting.
Distinctness: no duplicates in a set
A very important detail: in a set, elements are distinct. This means:
- We do not count duplicates.
- If something appears more than once in a description, we still treat it as only one element.
For example, consider:
- A description: {1, 2, 2, 3}
- As a set, this has the elements 1, 2, and 3 only.
So the set {1, 2, 2, 3} is really the same set as {1, 2, 3}.
The idea of “how many elements a set has” depends on distinct elements, not on repeated listings.
Order does not matter in a set
Another key feature: the order in which we list elements does not matter for a set.
For example, the following all describe the same set of numbers:
- {1, 2, 3}
- {3, 2, 1}
- {2, 1, 3}
All three are just the set whose elements are 1, 2, and 3.
This is different from ordered structures like lists or sequences, where order matters. For sets:
- We care which elements are there,
- We do not care in what order they are written.
Typical kinds of sets
As you move through mathematics, certain types of sets appear over and over. Here are a few simple examples to see the variety.
- Finite sets
These have a limited number of elements. For instance: - {red, blue, green}
- {1, 2, 3, 4, 5}
- {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
- Infinite sets
These go on without end. You cannot list all their elements one by one, but you can describe them.
Examples: - The set of all whole numbers: {0, 1, 2, 3, 4, 5, …}
- The set of all even numbers: {…, -4, -2, 0, 2, 4, 6, …}
- Sets defined by a property
Instead of listing all elements, we can describe a set by a rule or condition: - “The set of all people taller than 2 meters”
- “The set of all integers that are multiples of 5”
- “The set of all students in this class”
Here, “being taller than 2 meters” or “being a multiple of 5” is the property that decides membership.
In later chapters, you will see formal ways to write such descriptive definitions; for now, just notice that a set is often described by what all its elements have in common.
Examples and non-examples
Understanding what is and is not a set helps clarify the idea.
Clear sets
These have a well-defined rule for membership.
- “The set of months in the year”
- We know exactly which months those are.
- “The set of even numbers”
- We can tell for any number whether it is even or not.
- “The set of letters in the word ‘MATH’”
- The elements are M, A, T, H.
Vague or unclear collections
These are not good examples of sets in mathematics, because membership is not clearly defined.
- “The set of beautiful paintings”
- “Beautiful” is subjective; different people may disagree.
- “The set of interesting numbers”
- “Interesting” is not a precise property.
In mathematics, for something to be used as a set, we need a clear rule to decide if something is an element or not.
The idea of emptiness: the empty set
There is one special set worth mentioning early: the set with no elements at all. Think of a box with nothing inside.
This is called the empty set. It is still a set, just a set that does not have any elements.
- Every question “Is this in the empty set?” has the answer “No,”
because there is nothing in it.
The empty set will be important later when we do more with set notation and logic.
Thinking with sets
To get used to sets, it helps to view common situations in terms of “elements” and “sets”:
- A class of students can be seen as a set whose elements are the students.
- The set of books on your shelf has the individual books as elements.
- The set of all solutions to a particular equation (which you will meet later in algebra) has the solutions as its elements.
When you look at a situation, you can often ask:
- What is the set here?
- What are its elements?
- How is membership decided?
These questions help you translate everyday situations into mathematical language, which will be a key skill as you move through this course.
In later chapters, you will learn:
- How to write sets more formally (set notation),
- How to represent sets with pictures (Venn diagrams),
- How sets relate to logical ideas (logical operators and statements).
Here, it is enough to be comfortable with the basic picture: a set is a collection, and its elements are the things in that collection.