# 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 , that takes an element of the Cartesian product to the value .^{[1]}

- A function that sends an element
*x*to its equivalence class under a specified equivalence relation*E*,^{[2]}or, equivalently, a surjection from a set to another set.^{[3]}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.