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