Set (mathematics)

A set is a group of different things. These things are called elements or members of the set. They can be numbers, symbols, points in space, lines, geometric shapes, variables, functions, or even other sets.

In mathematics, sets are like building blocks. We describe how they work using rules called axioms. Almost every math idea can be explained using sets.

Set theory is a part of math that studies these rules. Most mathematicians today use a system called ZFC. This stands for Zermelo–Fraenkel set theory together with the axiom of choice. Sets help us organize and understand many parts of math.

Context

Before the late 1800s, mathematicians did not study sets closely and often confused them with sequences. They usually thought of infinity as something that could only be reached through a never-ending process.

The serious study of sets began with Georg Cantor. His work led to surprising results, like the fact that the number line has more points than the natural numbers.

Today, sets are used everywhere in mathematics. They help define structures in algebra and spaces in geometry. For example, we can say that the set of prime numbers never ends.

Main articles: Set theory, Naive set theory, Axiomatic set theory, Zermelo–Fraenkel set theory

Basic notions

In mathematics, a set is a group of different things, called elements or members. Sets can be made of numbers, shapes, or even other sets. You can describe a set by listing its elements or by giving a rule that all the elements follow, like the set of all prime numbers.

If a value is part of a set, we say it belongs to that set. For example, the number -3 belongs to the set of all integers, but 1.5 does not. There is also a special set with no elements at all, called the empty set. This set is very important in math!

Specifying a set

To describe a set, we can list its parts or give a rule that tells which items belong to the set.

One way to write a set is by listing its parts inside curly braces, like {1, 2, 3}. This is called roster notation. For example, {blue, white, red} is a set with the colors blue, white, and red. If we have a pattern, we can use an ellipsis (…); for example, {1, 2, 3, …, 10} means the numbers from 1 to 10.

Another way is called set-builder notation. Here we write a rule inside curly braces. For example, {n | n is an integer, and 0 ≤ n ≤ 19} means all whole numbers from 0 to 19. The vertical bar “|” reads as “such that.”

Main article: Set-builder notation

Subsets

Main article: Subset

A subset is a way to show a relationship between two sets. If we have two sets, we call them A and B. Then, A is a subset of B if every part in A is also a part in B.

For example, the set of all dogs is a subset of the set of all animals because every dog is an animal. In math, we can show this in a few ways: by saying "A is a subset of B," using the symbol ⊆ (like A ⊆ B), or saying "B contains A."

Basic operations

There are several ways to create new sets from existing sets, similar to how we add or multiply numbers. We often show these operations using special diagrams called Euler diagrams and Venn diagrams.

Intersection

The intersection of two sets A and B is a new set with only the elements that are in both A and B. We write this as A ∩ B. For example, if A has the numbers {1, 2, 3} and B has {2, 3, 4}, their intersection is {2, 3}.

Union

The union of two sets A and B is a new set with all the elements that are in A, B, or both. We write this as A ∪ B. Using the same example, the union of A and B is {1, 2, 3, 4}.

Set difference

Set difference happens when we take all elements from set A that are not in set B. We write this as A \ B. If A is {1, 2, 3} and B is {3, 4}, then A \ B is {1, 2}.

These operations help us understand relationships between different groups of things.

Functions

Main article: Function (mathematics)

A function is a rule that pairs each item in one set with one item in another set. For example, the square function takes a number and gives its square. So, the number 3 becomes 9, and the number -4 becomes 16.

We write a function using symbols like this: f: A → B. Here, A is the set of inputs, and B is the set of possible outputs. When we use the function f on a number a in A, we write f(a) to show the result.

External operations

In basic set operations, all elements come from sets we already know. This section looks at operations that make new sets with elements we haven’t considered before. These include Cartesian product, disjoint union, set exponentiation, and power set.

Cartesian product

The Cartesian product joins two sets into pairs. For sets A and B, their product A × B has every possible pair (a, b) where a is from A and b is from B. We can do this with three or more sets too, making triples or bigger groups.

Set exponentiation

Set exponentiation uses one set as the base and another as the exponent. For sets E and F, F^E has all possible ways to match each element of E to an element of F.

Power set

The power set of a set E has every possible group of elements from E. This includes the empty set and the set E itself. For example, the power set of {1, 2, 3} has eight parts: the empty set, each single number, each pair of numbers, and the whole set.

Disjoint union

The disjoint union of sets treats matching elements as different by labeling them with their original set. For sets A and B, A ⊔ B pairs each element with its set name, so there are no duplicates even if A and B share elements.

Cardinality

Main articles: Cardinality and Cardinal number

The cardinality of a set is how many elements it has. For a set with a small number of elements, this is like counting them. For example, the set {apple, banana, cherry} has a cardinality of 3.

Larger sets, even ones with infinitely many elements, can also be compared in size. Two sets have the same cardinality if we can match each element of one set to a unique element of the other set. For example, the set of even numbers and the set of all natural numbers have the same cardinality because we can match each even number to half of itself, like 2 to 1, 4 to 2, and so on.

Axiom of choice

Main article: Axiom of choice

The axiom of choice is a rule in math that helps solve problems with groups of sets. It says that if you have many non-empty sets, you can pick one item from each set all at once. This idea is useful because it helps mathematicians work with very large groups in a clear way.

Zorn's lemma

Main article: Zorn's lemma

Zorn's lemma works together with the axiom of choice. It helps prove that some groups have a "largest" or "most complete" member. For example, it can show that every space made of vectors has a basis — a small group of vectors that can create every other vector in the space.

Transfinite induction

Main articles: Well-order and Transfinite induction

Transfinite induction is like the usual mathematical induction, but it works for more than just counting numbers. It lets us prove something is true for every item in a well-ordered set — a group where every part has a smallest piece. This method helps us understand special numbers in math called ordinals and cardinals.