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 jth projection map, written\( {\displaystyle \mathrm {proj} _{j}} \), 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.