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 $\mathrm {proj} _{j}\!$ , that takes an element ${\vec {x}}=(x_{1},\ \ldots ,\ x_{j},\ \ldots ,\ x_{k})$ of the Cartesian product $(X_{1}\times \cdots \times X_{j}\times \cdots \times X_{k})$ to the value $\mathrm {proj} _{j}({\vec {x}})=x_{j}$ .^[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.