Set Theory

Set Theory explores the fundamental concepts of sets, including operations, relations, and applications in mathematics.

Advanced Topics

Functions as Special Relations

What is a Function?

A function is a special type of relation that connects each element of one set (the domain) to exactly one element of another set (the codomain).

Notation and Language

If \(f\) is a function from \(A\) to \(B\), we write \(f: A \to B\). For each \(a \in A\), \(f(a)\) gives an element in \(B\).

Types of Functions

  • Injective (One-to-One): Different inputs map to different outputs.
  • Surjective (Onto): Every output in the codomain is used.
  • Bijective: Both injective and surjective.

Functions in Action

Functions can describe everything from how a vending machine gives snacks to how a computer translates code!

Examples

  • The function \(f(x) = x^2\) maps numbers to their squares.

  • Assigning each student to their locker number is a function.

In a Nutshell

A function matches each input to exactly one output—no repeats, no leftovers.