Set facts for kids

Georg Cantor, in 1894. Cantor was the first mathematician to talk about sets

Cantor's original definition of a set

A set is a group of things in mathematics. These things are called members or elements. A set is special because it's defined only by what's inside it. If two sets have the exact same members, they are considered the same set.

Example of a set of polygons

A set cannot have the same member more than once. The order of the members does not matter. Anything can be a member of a set. This even includes other sets!

What Can You Do with Sets?

Imagine a set is like a bag. You can put different things into this bag.

Checking for Members: "Element Of"

You might want to know if a certain thing is inside your set (or bag). Mathematicians call this "element of." If something is in the set, it is an element of that set.

The symbol for "element of" is $\in$ . So, $a \in \mathit{A}$ means that $a$ is in the set $\mathit{A}$ .

Unlike a real bag, a set only cares if an item is there or not. It doesn't matter if you have one orange or ten oranges. If an orange is in the set, it's just "an orange."

The Empty Set: An Empty Bag

Just like a bag can be empty, a set can also be empty. The empty set has nothing in it. It is also called the null set. The symbol for the empty set is $\varnothing$ or $\{\}$ .

The Universe: All Things to Consider

Sometimes, when we talk about sets, we need to think about a bigger group that contains all the possible things we are interested in. This bigger group is called the universe.

For example, if you are looking at sets of different types of American cars, the universe would be the set of all American cars. The universe is usually shown with the letter $U$ .

Comparing Sets: Are They the Same?

You can compare two sets to see if they are equal. This is like looking into two different bags. If they contain the exact same things, they are equal. The order of the items does not matter.

For example: If set $\mathit{A}=\{ Stanford, Stanley \}$ and set $\mathit{B}=\{ Stanley, Stanford \}$ , these two sets are the same.

How Big Is a Set? Cardinality

Mathematicians often want to know how many elements are in a set. This is called the cardinality of the set. For sets with a limited number of items (called finite sets), the cardinality is simply a number.

The empty set has a cardinality of 0.
The set $\{ apple, orange \}$ has a cardinality of 2.

Two sets have the same cardinality if you can match up their elements perfectly. This means you can pair each element from one set with an element from the other set, with no elements left over.

For example: The set $\{ apple, orange \}$ and the set $\{ sun, moon \}$ have the same cardinality. You could pair "apple" with "sun" and "orange" with "moon."

Infinite Cardinality: Sets That Never End

Sometimes, a set has an infinite number of elements. This means its cardinality is not a simple number. The set of all integers (whole numbers) is an example of a set with infinite cardinality.

Some infinite sets are "bigger" than others. For instance, there are more real numbers than there are natural numbers. This means you cannot perfectly pair up all natural numbers with all real numbers, even if you tried forever.

Countable and Uncountable Sets

If you can count the elements of a set, even if it takes forever, it's called a countable set. All finite sets are countable. Some infinite sets are also countable, like the natural numbers (Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {1, 2, 3, ...} ). We call natural numbers "counting numbers" because we use them to count.

An uncountable set is an infinite set that is impossible to count. No matter how you try to count its elements, you will always miss some. The set of all real numbers is an uncountable set. Even a small range of numbers, like from 0 to 1, can be uncountable!

Subsets: Sets Inside Other Sets

A subset of regular polygons

Look at set $A=\{ a, b \}$ and set $B=\{ a, b, c, d \}$ . You can see that every element in set $A$ is also in set $B$ . When this happens, we say that $A$ is a subset of $B$ . It is written like this: $\{a,b\} \subseteq \{a,b,c,d\}$ .

In general, if all elements of set $A$ are also elements of set $B$ , we write: $A \subseteq B$ . This is usually read as " $A$ is contained in $B$ ."

Example: Every Chevrolet is an American car. So, the set of all Chevrolets is a subset of the set of all American cars.

How to Combine Sets: Set Operations

Intersection of two sets of polygons

There are different ways to combine sets to create new ones.

Intersections: What Sets Have in Common

The intersection of two sets, $A$ and $B$ , is a new set. It contains all the elements that are in set $A$ and also in set $B$ at the same time. The symbol for intersection is $\cap$ , so we write $A \cap B$ .

Example: If $A$ is the set of all cheap cars, and $B$ is the set of all American cars, then $A \cap B$ is the set of all cheap American cars.

Unions: All Elements Together

Union of two sets of polygons

The union of two sets, $A$ and $B$ , is a new set. It contains all the elements that are in set $A$ or in set $B$ . This "or" includes elements that are in both sets. The symbol for union is $\cup$ , so we write $A \cup B$ .

It's interesting to note that the intersection is always a subset of the union: $(A \cap B) \subseteq (A \cup B)$ .

Example: If $A$ is the set of all cheap cars, and $B$ is the set of all American cars, then $A \cup B$ is the set of all cars that are either cheap, or American, or both. It does not include expensive cars that are not from America.

Complements: What's Left Over

"Complement" can mean two slightly different things in set theory.

The complement of $A$ is the universe $U$ without any of the elements of $A$ .

We write this as $A^{\rm C} = U \setminus A$ . Example: If $U$ is the set of all cars, and $A$ is the set of all cheap cars, then $A$ ^C is the set of all expensive cars.

Differences of two sets of polygons

The set difference of $A$ and $B$ is the set $B$ without any of the elements of $A$ .

We write this as $B \setminus A$ . It's also called the relative complement of $A$ in $B$ . Example: If $A$ is the set of all cheap cars, and $B$ is the set of all American cars, then $B \setminus A$ is the set of all expensive American cars.

If you switch the sets in a set difference, the result changes. For example, $A \setminus B$ would be the set of all cheap cars that are not made in America.

How to Write About Sets

Most mathematicians use uppercase italic letters for sets (like $A$ , $B$ , $C$ ). The elements inside the sets are usually written with lowercase letters.

One way to show a set is by listing its members. You separate them with commas and put them inside curly braces. For example:

$X=\{1, 2, 3\}$ is a set with members 1, 2, and 3.

Another way is called "set-builder notation." This describes what is true about the members of the set:

{x | x is a natural number & x < 4}.

This reads: "the set of all x such that x is a natural number and x is less than four." The symbol "|" means "such that."

The empty set can be written as $\empty$ , $\varnothing$ , or $\{\}$ .

When an object a is a member of set $A$ , it is written as:

$a \in A$ .

This reads: "a is a member of $A$ ."

Visualizing Sets: Venn Diagrams

Intersection

A \cap B

To help understand how sets work, mathematicians use Venn diagrams. These diagrams use circles to show individual sets. The universe (all things being considered) is shown as a rectangle around the circles. The results of set operations are shown by coloring certain areas. In the picture, the left circle is set $A$ and the right circle is set $B$ . The colored part shows their intersection.

Important Special Sets

Some sets are used very often in mathematics. They even have special symbols. Many of these special sets are written using a special bold font called blackboard bold.

$\mathbb{P}$ : This is the set of all prime numbers (numbers greater than 1 that can only be divided by 1 and themselves, like 2, 3, 5, 7...).
$\mathbb{N}$ : This is the set of all natural numbers. These are the numbers you use for counting: {1, 2, 3, ...}. Sometimes, it includes 0: {0, 1, 2, 3, ...}.
$\mathbb{Z}$ : This is the set of all integers (whole numbers, including positive, negative, and zero): {..., -2, -1, 0, 1, 2, ...}.
$\mathbb{Q}$ : This is the set of all rational numbers. These are numbers that can be written as a fraction Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{matrix}\frac{a}{b} \end{matrix} , where a and b are integers and b is not zero. For example, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{matrix} \frac{1}{4} \end{matrix} and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{matrix}\frac{11}{6} \end{matrix} are rational numbers. All integers are rational because you can write any integer a as $\begin{matrix} \frac{a}{1} \end{matrix}$ .
$\mathbb{R}$ : This is the set of all real numbers. This includes all rational numbers, plus irrational numbers (numbers that cannot be written as fractions, like $\pi,$ $e,$ and √2).
$\mathbb{C}$ : This is the set of all complex numbers.

Each of these sets has an infinite number of elements. They also fit inside each other like Russian dolls: $\mathbb{P} \subset \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$ .

Fun with Paradoxes: The Barber Problem

The mathematician Bertrand Russell found some tricky problems with how sets were first defined. One famous problem is called Russell's paradox. A simpler version, which is easier to understand, is called the Barber paradox.

The Barber Paradox

Imagine a small town with a barber. In this town, all the men either shave themselves, or they go to the barber shop to be shaved by the barber.

The barber has a rule: "The barber shaves all men who do not shave themselves. He only shaves those men."

Now, think about the barber himself: Who shaves the barber?

If the barber shaves himself, then according to his own rule, he cannot be the barber (because a barber only shaves those who do not shave themselves).
If the barber does not shave himself, then he falls into the group of men who do not shave themselves. According to his rule, he must shave himself.

This creates a puzzle! It shows how careful mathematicians need to be when defining things like sets.

The following books explore sets in more detail:

Halmos, Paul R., Naive Set Theory, Princeton, N.J.: Van Nostrand (1960) ISBN: 0-387-90092-6
Stoll, Robert R., Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN: 0-486-63829-4
Allenby, R.B.J.T, Rings, Fields and Groups, Leeds, England: Butterworth Heinemann (1991) ISBN: 0-340-54440-6