Set (mathematics)

In mathematics, a set is a collection of different things. These things are called elements or members of the set. They can be many types of objects, such as numbers, symbols, points in space, lines, geometric shapes, variables, functions, or even other sets.

Mathematics usually doesn’t give a strict definition of what a "set" is, because doing so would need to use other ideas that are not yet explained. Instead, sets are like basic building blocks. Their properties are described using rules called axioms, based on our natural idea of grouping things together. Almost every other mathematical object can be carefully described using sets.

Set theory is the area of mathematics that looks at different sets of rules and what they lead to. Since the middle of the 1900s, most mathematicians have used a system called ZFC, which stands for Zermelo–Fraenkel set theory together with the axiom of choice. Sets are important because they help organize and understand many different 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, and were not ready to think about infinite sets as real collections.

The serious study of infinite sets began with Georg Cantor, who lived from 1845 to 1918. His work led to some surprising results, like the fact that the number line contains more points than the natural numbers, and that any piece of a line has just as many points as the whole line. Thinking about a set that includes every possible set created a problem called Russell's paradox, which caused a big debate in mathematics about how to properly define sets.

Today, sets are used everywhere in mathematics. They help define structures in algebra and spaces in geometry, and many old theorems are now expressed using sets. For example, we can say that the set of prime numbers is endless.

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

Basic notions

In mathematics, a set is a collection 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 elements or give a rule that tells us which items belong to the set.

One way to write a set is by listing its elements inside curly braces, like {1, 2, 3}. This is called roster notation. For example, {blue, white, red} is a set containing 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 special kind of relationship between two sets. If we have two sets, let's call them A and B, then A is a subset of B if every element in A is also an element 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 relationship 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 adding or multiplying numbers works. 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 that contains 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 that contains 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 element of one set with a unique element of another set. For example, the square function takes any real 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 apply the function f to a number a in A, we write f(a) to show the result.

External operations

In basic set operations, all elements come from previously defined sets. This section explores operations that create new sets with elements outside those already considered. These include Cartesian product, disjoint union, set exponentiation, and power set.

Cartesian product

The Cartesian product combines two sets into pairs. For sets A and B, their product A × B includes every possible pair (a, b) where a is from A and b is from B. This can be extended to three or more sets, creating triples or larger groups.

Set exponentiation

Set exponentiation involves one set acting as the base and another as the exponent. For sets E and F, F^E consists of all possible functions that map elements from E to F.

Power set

The power set of a set E includes every possible subset of E, including the empty set and E itself. For example, the power set of {1, 2, 3} has eight elements: the empty set, each single number, each pair of numbers, and the full set.

Disjoint union

The disjoint union of sets treats overlapping elements as distinct by labeling them with their original set. For sets A and B, A ⊔ B pairs each element with its set identifier, ensuring 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 just 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 have their size compared. 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 concept in mathematics that helps solve problems involving collections of sets. It states that, given any group of non-empty sets, we can pick one item from each set all at once. This idea is important because it lets mathematicians work with infinite collections in a clear way. There are different ways to express this axiom that are equally true and useful in proofs.

Zorn's lemma

Main article: Zorn's lemma

Zorn's lemma is a tool that works hand-in-hand with the axiom of choice. It helps prove that certain collections have a "largest" or "most complete" member. For example, it can show that every space made of vectors has a basis — a small set of vectors that can build every other vector in the space through combinations.

Transfinite induction

Main articles: Well-order and Transfinite induction

Transfinite induction is like the usual method of 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 collection where every part has a smallest piece. This method is key to understanding special numbers in math called ordinals and cardinals.