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 $$\mathrm{proj}_j(\vec{x}) = x_j.$$
 * 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.