## Sets and Functions: A Brief Overview

### Sets

A set is a collection of distinct objects or elements. These elements can be anything, from numbers and letters to more complex objects.

- Notation: Sets are usually denoted by curly braces {}. For example, the set of even numbers between 1 and 10 can be written as {2, 4, 6, 8}.
- Elements: The individual objects in a set are called elements.
- Subset: A set A is a subset of set B if all elements of A are also elements of B. This is denoted as A ⊆ B.
- Union: The union of sets A and B is the set of all elements that are in A or B or both. It's denoted as A ∪ B.
- Intersection: The intersection of sets A and B is the set of all elements that are in both A and B. It's denoted as A ∩ B.
- Complement: The complement of set A (relative to a universal set U) is the set of all elements in U that are not in A. It's denoted as A'.

### Functions

A function is a mathematical rule that associates each element in a set (the domain) with a unique element in another set (the codomain).

- Domain: The set of inputs for a function.
- Codomain: The set of possible outputs for a function.
- Range: The subset of the codomain that actually contains the outputs of the function.
- One-to-one: A function is one-to-one if no two different inputs produce the same output.
- Onto: A function is onto if every element in the codomain has at least one corresponding element in the domain.
- Inverse function: If a function is both one-to-one and onto, it has an inverse function. The inverse function "undoes" the original function.

Example: The function f(x) = 2x + 1 maps elements from the set of real numbers (domain) to the set of real numbers (codomain). For example, f(3) = 7. This function is both one-to-one and onto, so it has an inverse function.

Key Concepts:

- Set operations: union, intersection, complement
- Function properties: one-to-one, onto
- Inverse functions