Map

A map (or mapping) is another term for a function between sets.

Definition

A map $f: X \to Y$ is a rule that assigns to each element $x \in X$ exactly one element $f(x) \in Y$.

• $X$ is the domain
• $Y$ is the codomain
• $f(X) = \{f(x) : x \in X\}$ is the image (or range)

Terminology

The terms “map,” “mapping,” and “function” are often used interchangeably. Some conventions:

• Map/Mapping: Often preferred in algebra and topology
• Function: Often preferred in analysis
• Morphism: Used in category theory for structure-preserving maps

Types of Maps

• injection (one-to-one): $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$
• surjection (onto): $f(X) = Y$
• bijection (one-to-one and onto): both injective and surjective

Related

• function
• domain
• codomain
• image