What is a set in mathematics?

A set in mathematics is a collection of distinct objects, considered as an entity.

In mathematics, a set is essentially a group of items, known as elements or members, that are grouped together because they share a common property or simply because they are listed together. These elements can be anything: numbers, letters, symbols, or even other sets. For example, the set of vowels in the English alphabet can be written as {a, e, i, o, u}.

Sets are usually denoted by curly braces {}. If an element belongs to a set, we use the symbol ∈. For instance, if we have a set A = {1, 2, 3}, we can say 1 ∈ A, meaning 1 is an element of set A. Conversely, if an element is not in the set, we use the symbol ∉. So, 4 ∉ A means 4 is not an element of set A.

There are different types of sets. A finite set has a limited number of elements, like {1, 2, 3}. An infinite set has an unlimited number of elements, such as the set of all natural numbers {1, 2, 3, ...}. There are also empty sets, which contain no elements at all and are denoted by {} or ∅.

Understanding sets is fundamental in mathematics because they form the basis for more complex concepts like functions, sequences, and probability. By learning about sets, you gain a better grasp of how to organise and analyse groups of objects, which is a crucial skill in many areas of maths and science.