Introduction to Sets (2024)

Forget everything you know about numbers.

In fact, forget you even know what a number is.

This is where mathematics starts.

Instead of math with numbers, we will now think about math with "things".

Definition

What is a set? Well, simply put, it's a collection.

First we specify a common property among "things" (we define this word later) and then we gather up all the "things" that have this common property.

Introduction to Sets (1)

For example, the items you wear: hat, shirt, jacket, pants, and so on.

I'm sure you could come up with at least a hundred.

This is known as a set.

Or another example is types of fingers.

This set includes index, middle, ring, and pinky.

Introduction to Sets (2)

So it is just things grouped together with a certain property in common.

Notation

There is a fairly simple notation for sets. We simply list each element (or "member") separated by a comma, and then put some curly brackets around the whole thing:

Introduction to Sets (3)

The curly brackets { } are sometimes called "set brackets" or "braces".

This is the notation for the two previous examples:

{socks, shoes, watches, shirts, ...}
{index, middle, ring, pinky}

Notice how the first example has the "..." (three dots together).

The three dots ... are called an ellipsis, and mean "continue on".

So that means the first example continues on ... for infinity.

(OK, there isn't really an infinite amount of things you could wear, but I'm not entirely sure about that! After an hour of thinking of different things, I'm still not sure. So let's just say it is infinite for this example.)

So:

  • The first set {socks, shoes, watches, shirts, ...} we call an infinite set,
  • the second set {index, middle, ring, pinky} we call a finite set.

But sometimes the "..." can be used in the middle to save writing long lists:

Example: the set of letters:

{a, b, c, ..., x, y, z}

In this case it is a finite set (there are only 26 letters, right?)

Numerical Sets

So what does this have to do with mathematics? When we define a set, all we have to specify is a common characteristic. Who says we can't do so with numbers?

Set of even numbers: {..., −4, −2, 0, 2, 4, ...}
Set of odd numbers: {..., −3, −1, 1, 3, ...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}

And so on. We can come up with all different types of sets.

We can also define a set by its properties, such as {x|x>0} which means "the set of all x's, such that x is greater than 0", see Set-Builder Notation to learn more.

And we can have sets of numbers that have no common property, they are just defined that way. For example:

{2, 3, 6, 828, 3839, 8827}
{4, 5, 6, 10, 21}
{2, 949, 48282, 42882959, 119484203}

Are all sets that I just randomly banged on my keyboard to produce.

Why are Sets Important?

Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pretty pointless. But it's only when we apply sets in different situations do they become the powerful building block of mathematics that they are.

Math can get amazingly complicated quite fast. Graph Theory, Abstract Algebra, Real Analysis, Complex Analysis, Linear Algebra, Number Theory, and the list goes on. But there is one thing that all of these share in common: Sets.

Universal Set

Introduction to Sets (4)

At the start we used the word "things" in quotes.

We call this the universal set. It's a set that contains everything. Well, not exactly everything. Everything that is relevant to our question.

Introduction to Sets (5)

In Number Theory the universal set is all the integers, as Number Theory is simply the study of integers.

Introduction to Sets (6)

But in Calculus (also known as real analysis), the universal set is almost always the real numbers.

Introduction to Sets (7)And in complex analysis, you guessed it, the universal set is the complex numbers.

Some More Notation

Introduction to Sets (8) When talking about sets, it is fairly standard to use Capital Letters to represent the set, and lowercase letters to represent an element in that set.

So for example, A is a set, and a is an element in A. Same with B and b, and C and c.

Now you don't have to listen to the standard, you can use something like m to represent a set without breaking any mathematical laws (watch out, you can get π years in math jail for dividing by 0), but this notation is pretty nice and easy to follow, so why not?

Also, when we say an element a is in a set A, we use the symbol Introduction to Sets (9) to show it.
And if something is not in a set use Introduction to Sets (10).

Example: Set A is {1,2,3}. We can see that 1 Introduction to Sets (11) A, but 5 Introduction to Sets (12) A

Equality

Two sets are equal if they have precisely the same members. Now, at first glance they may not seem equal, so we may have to examine them closely!

Example: Are A and B equal where:

  • A is the set whose members are the first four positive whole numbers
  • B = {4, 2, 1, 3}

Let's check. They both contain 1. They both contain 2. And 3, And 4. And we have checked every element of both sets, so: Yes, they are equal!

And the equals sign (=) is used to show equality, so we write:

A = B

Example: Are these sets equal?

  • A is {1, 2, 3}
  • B is {3, 1, 2}

Yes, they are equal!

They both contain exactly the members 1, 2 and 3.

It doesn't matter where each member appears, so long as it is there.

Introduction to Sets (13)

Subsets

When we define a set, if we take pieces of that set, we can form what is called a subset.

Example: the set {1, 2, 3, 4, 5}

A subset of this is {1, 2, 3}. Another subset is {3, 4} or even another is {1}, etc.

But {1, 6} is not a subset, since it has an element (6) which is not in the parent set.

In general:

A is a subset of B if and only if every element of A is in B.

So let's use this definition in some examples.

Example: Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?

1 is in A, and 1 is in B as well. So far so good.

3 is in A and 3 is also in B.

4 is in A, and 4 is in B.

That's all the elements of A, and every single one is in B, so we're done.

Yes, A is a subset of B

Note that 2 is in B, but 2 is not in A. But remember, that doesn't matter, we only look at the elements in A.

Let's try a harder example.

Example: Let A be all multiples of 4 and B be all multiples of 2.
Is A a subset of B? And is B a subset of A?

Well, we can't check every element in these sets, because they have an infinite number of elements. So we need to get an idea of what the elements look like in each, and then compare them.

The sets are:

  • A = {..., −8, −4, 0, 4, 8, ...}
  • B = {..., −8, −6, −4, −2, 0, 2, 4, 6, 8, ...}

By pairing off members of the two sets, we can see that every member of A is also a member of B, but not every member of B is a member of A:

Introduction to Sets (14)

So:

A is a subset of B, but B is not a subset of A

Proper Subsets

If we look at the defintion of subsets and let our mind wander a bit, we come to a weird conclusion.

Let A be a set. Is every element of A in A?

Well, umm, yes of course, right?

So that means that A is a subset of A. It is a subset of itself!

This doesn't seem very proper, does it? If we want our subsets to be proper we introduce (what else but) proper subsets:

A is a proper subset of B if and only if every element of A is also in B, and there exists at least one element in B that is not in A.

This little piece at the end is there to make sure that A is not a proper subset of itself: we say that B must have at least one extra element.

Example:

{1, 2, 3} is a subset of {1, 2, 3}, but is not a proper subset of {1, 2, 3}.

Example:

{1, 2, 3} is a proper subset of {1, 2, 3, 4} because the element 4 is not in the first set.

Notice that when A is a proper subset of B then it is also a subset of B.

Even More Notation

When we say that A is a subset of B, we write A ⊆ B

Or we can say that A is not a subset of B by A ⊈ B

When we talk about proper subsets, we take out the line underneath and so it becomes A ⊂ B or if we want to say the opposite A ⊄ B

Empty (or Null) Set

This is probably the weirdest thing about sets.

Introduction to Sets (15)

As an example, think of the set of piano keys on a guitar.

"But wait!" you say, "There are no piano keys on a guitar!"

And right you are. It is a set with no elements.

This is known as the Empty Set (or Null Set).There aren't any elements in it. Not one. Zero.

It is represented by

Or by {} (a set with no elements)

Some other examples of the empty set are the set of countries south of the south pole.

So what's so weird about the empty set? Well, that part comes next.

Empty Set and Subsets

So let's go back to our definition of subsets. We have a set A. We won't define it any more than that, it could be any set. Is the empty set a subset of A?

Going back to our definition of subsets, if every element in the empty set is also in A, then the empty set is a subset of A. But what if we have no elements?

It takes an introduction to logic to understand this, but this statement is one that is "vacuously" or "trivially" true.

A good way to think about it is: we can't find any elements in the empty set that aren't in A, so it must be that all elements in the empty set are in A.

So the answer to the posed question is a resounding yes.

The empty set is a subset of every set, including the empty set itself.

Order

No, not the order of the elements. In sets it does not matter what order the elements are in.

Example: {1,2,3,4} is the same set as {3,1,4,2}

When we say order in sets we mean the size of the set.

Another (better) name for this is cardinality.

A finite set has finite order (or cardinality). An infinite set has infinite order (or cardinality).

For finite sets the order (or cardinality) is the number of elements.

Example: {10, 20, 30, 40} has an order of 4.

For infinite sets, all we can say is that the order is infinite. Oddly enough, we can say with sets that some infinities are larger than others, but this is a more advanced topic in sets.

Arg! Not more notation!

Nah, just kidding. No more notation.

by

Ricky Shadrach

and

Rod Pierce

366, 367, 368, 1051, 1052, 1053, 9070, 2424, 2425, 2426

Activity: Subsets

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Introduction to Sets (2024)

FAQs

What is the introduction to sets? ›

Sets - An Introduction

A set is a collection of objects. The objects in a set are called its elements or members. The elements in a set can be any types of objects, including sets! The members of a set do not even have to be of the same type.

What is the concept of sets? ›

A set is a collection of objects or groups of objects. These objects are often called elements or members of a set. For example, a group of players in a cricket team is a set. Since the number of players in a cricket team could be only 11 at a time, thus we can say, this set is a finite set.

How do you explain a set? ›

In mathematics, a set is defined as a collection of distinct, well-defined objects forming a group. There can be any number of items, be it a collection of whole numbers, months of a year, types of birds, and so on. Each item in the set is known as an element of the set. We use curly brackets while writing a set.

What are the basic ideas of sets math? ›

  • Intuitively, a set is a collection of objects with certain properties. ...
  • We say that a set A is a subset of a set B if every element of A is also an element of B, and write A⊂B or B⊃A.
  • Two sets are equal if they contain the same elements. ...
  • and similar definitions for (a,b), [a,b), (−∞,b], and (−∞,b).
Sep 5, 2021

What are the rules of sets? ›

Algebra of Sets
Idempotent Laws(a) A ∪ A = A(b) A ∩ A = A
Associative Laws(a) (A ∪ B) ∪ C = A ∪ (B ∪ C)(b) (A ∩ B) ∩ C = A ∩ (B ∩ C)
Commutative Laws(a) A ∪ B = B ∪ A(b) A ∩ B = B ∩ A
Distributive Laws(a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)(b) A ∩ (B ∪ C) =(A ∩ B) ∪ (A ∩ C)
De Morgan's Laws(a) (A ∪B)c=Ac∩ Bc(b) (A ∩B)c=Ac∪ Bc
3 more rows

What is the formula for sets? ›

The general set formula is expressed as n(A∪B) = n(A) + n(B) - n(A⋂B), where A and B represent two sets. Here, n(A∪B) denotes the count of elements existing in either set A or B, while n(A⋂B) indicates the count of elements shared by both sets A and B.

How do you introduce a set to students? ›

There are three ways to write or represent a set: statement form, roster form, and set-builder form. This is the simplest way to introduce the concept of a set and it's how elementary school students are introduced to the idea of grouping. Here, the rule of the elements is written as a well-defined description.

What is the purpose of sets? ›

Sets are used to store a collection of linked things. They are essential in all fields of mathematics because sets are used or referred to in some manner in every branch of mathematics. They are necessary for the construction of increasingly complicated mathematical structures.

What is an example of a set? ›

A set is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set in a curly bracket {…}. For example, set A is a collection of all the natural numbers, such as A = {1,2,3,4,5,6,7,8,…..∞}.

How to teach sets in maths? ›

Representation of Sets

The sets are represented in curly braces, {}. For example, {2,3,4} or {a,b,c} or {Bat, Ball, Wickets}. The elements in the sets are depicted in either the Statement form, Roster Form or Set Builder Form.

What are the three ways of defining sets? ›

The roster method lists all the elements or members in the set, whereas a description in words explains what elements are in the set using a sentence. And set-builder notation expresses how elements are given membership in the set by specifying the properties that define the collection of objects.

What is the basic set theory? ›

Sets are well-determined collections that are completely characterized by their elements. Thus, two sets are equal if and only if they have exactly the same elements. The basic relation in set theory is that of elementhood, or membership.

What is sets in your own words? ›

Sets in mathematics, are simply a collection of distinct objects forming a group. A set can have any group of items, be it a collection of numbers, days of a week, types of vehicles, and so on. Every item in the set is called an element of the set. Curly brackets are used while writing a set.

What is the set theory in simple terms? ›

In naive set theory, a set is a collection of objects (called members or elements) that is regarded as being a single object. To indicate that an object x is a member of a set A one writes x ∊ A, while x ∉ A indicates that x is not a member of A.

What is the study of sets? ›

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory — as a branch of mathematics — is mostly concerned with those that are relevant to mathematics as a whole.

What is describing of sets? ›

A set can be described two ways -- by roster method or by using set-builder notation. The roster method simply lists all the elements in the set. For example, set A could be described using braces like this: A = {1, 2, 3, 4, 5}.

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