A set is a collection of objects or elements, which can be anything, such as numbers, letters, or even other sets. The elements in a set are typically listed within curly braces, and each element is listed only once. For example, the set of all even numbers can be represented as {2, 4, 6, 8, …}. Sets are used to define and study many mathematical concepts, such as functions and groups.

A function is a rule that assigns a unique output value to each input value. Functions are used to describe relationships between different mathematical objects, such as numbers, sets, or even other functions. A function can be represented using a formula, a graph, or a table. For example, the function f(x) = x^2 assigns the output value x^2 to each input value x. Functions can be composed and combined using various operations, such as addition, subtraction, multiplication, and division.

A group is a set of elements together with an operation that combines two elements and produces another element in the group. The operation must satisfy certain properties, such as associativity, identity, and invertibility. Groups are used to study symmetry and structure in mathematics and science. Examples of groups include the group of integers under addition, the group of 2×2 matrices under matrix multiplication, and the group of symmetries of a regular polygon.

In summary, sets, functions, and groups are important concepts in mathematics that are used to study various mathematical structures and relationships. Sets provide a way to collect and organize mathematical objects, functions describe relationships between objects, and groups represent symmetries and structure in mathematics and science.