Projection (set theory)

In set theory, a projection is one of two closely related types of functions or operations, namely:

• A set-theoretic operation typified by the $j$^th projection map, written ${\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}_j,$ that takes an element $\overrightarrow{x}=(x_1,\dots ,x_j,\dots ,x_k)$ of the Cartesian product $(X_1\times \cdots \times X_j\times \cdots \times X_k)$ to the value ${\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}_j(\overrightarrow{x})=x_j.$
• A function that sends an element $x$ to its equivalence class under a specified equivalence relation $E,$ or, equivalently, a surjection from a set to another set. The function from elements to equivalence classes is a surjection, and every surjection corresponds to an equivalence relation under which two elements are equivalent when they have the same image. The result of the mapping is written as $[x]$ when $E$ is understood, or written as $[x{]}_E$ when it is necessary to make $E$ explicit.

See also

• Cartesian product – Mathematical set formed from two given sets
• Projection (mathematics) – Mapping equal to its square under mapping composition
• Projection (linear algebra) – Idempotent linear transformation from a vector space to itself
• Projection (relational algebra) – Operation that restricts a relation to a specified set of attributes
• Relation (mathematics) – Relationship between two sets, defined by a set of ordered pairs